College Algebra. This is episode 25 and I'm Dennis Allison. You know we've been talking about matrices, how you multiply matrices, find the inverse of a matrix and some of the applications for this sort of thing. Today we look at something that's related to that. Let's go to our list of objectives. Today we're going to introduce these two by two, three by three determinants and larger determinants. We'll also look at how determinants are related to deciding whether a matrix has an inverse, so determinants and inverse matrices. Then we'll look at the three elementary operations that we can perform on determinants. These are the same elementary operations that we performed when we were reducing matrices, although now we'll be applying them to both rows and columns. And finally we'll look at one of the one of the primary applications of determinants, not the only one, but Kramer's rule. And these things are all in your text. So let's look at first of all what is a determinant. Okay let's begin by tying this in with something we've already seen. You know normally for a matrix we use a capital letter to represent the matrix for example a capital A. And if this is a two by two matrix then the entries in a two by two matrix I might call A, B, C and D. You know there's another way of writing the matrix that is you can use parentheses. In your textbook you sometimes see parentheses used A, B, C and D. But what you never see here are vertical bars because that represents something else. In fact that's what, that's the notation that we use for a determinant. Now if I want to take the determinant of A the way I'll abbreviate that is I'll put vertical bars on the A's. Looks sort of like an absolute value but it doesn't mean that. Another way to abbreviate determinant of A is to write D, E, T parentheses A to mean the determinant of A. And then when I go to actually express the two by two arrangement I'll put vertical bars out here and I'll put A, B, C and D. Now the way I evaluate a determinant is I multiply the number in the first row first column times the number in the second row in the second column. A times D minus B times C. And that would be A, D minus B, C. In other words I take this product minus that product and I come up with a real number answer. You know for a matrix you never get a real number answer for these things. One matrix can equal another matrix but a matrix doesn't ever equal a real number. Not if it's like a two by two matrix but for a determinant it has a real number value. Let me just work an example right below here. Suppose I had the determinant 4, negative 2, negative 3 and 5. And I want to find the value of this determinant. Well I take the product 4 times 5 minus the product negative 2 times negative 3. Negative 2 times negative 3. Now that's going to give me 20 minus 6 is 14. So I say 14 is the value of this determinant. You know another representation for a matrix going back up to this first notation that you haven't, I don't know that we've used this much, this notation in class is another way I could represent matrix A is I could say that the entries are, I could use a little a for every entry, a one one, a one two, a two one, a two two. Now this time I'm putting two subscripts below or just below and to the right of each of each a and this represents first row first column, first row second column and then second row second column. So in other words the two numbers that you see in the subscript represent the row number and the column number of that entry. Now that would be equivalent to saying a, b, c, and d, but this way quickly I can identify which position the number came from. So in this case, well let's see for a determinant, if I were going to take the determinant of a, I would write this as a one one times a two two minus a one two times a two one. Okay let's work another example now of a two by two determinant and then we'll look at it from a slightly different point of view. So I have, I have two ways of representing the entries of a matrix and that gives me two ways of expressing the value of this determinant. Okay so another two by two example. Suppose this time I had matrix b and the entries of matrix b this time might be zero and four and negative three and negative one and I'd like to evaluate the determinant of matrix b which I'll write as the determinant zero four negative three negative one. Now once again I'll take this product minus this product so that it'll be zero times negative one minus four times negative 13 and I get zero plus twelve is twelve. Okay so that's said to be the value of this determinant. The matrix had no value but the determinant does. Okay now that's sort of the, sort of the quick way of evaluating a determinant, a two by two determinant. Now let me show you another way, another definition for evaluating it and this definition can be expanded to larger determinants. You know one thing I haven't mentioned yet so far is that when you take the determinant of a matrix the matrix has to be square like a two by two or a three by three. So if I were to give you a two rows, three columns you couldn't take a determinant of that. Okay so going back to the two by two case suppose I want to take the determinant, this time I'll use this notation. I want to take the determinant of a matrix C and let's say the entries of the determinant this time are five, two, three and four. Five, two, three and four. Now I want to look at this from a slightly different point of view. The number in the very first position is said to be in a positive position. It has nothing to do with the fact that five is positive. We just say that the number in the first row, first column is in a positive position. The number in the first row, second column is in a negative position. So positive and negative and the signs alternate going down as well. Five's in a positive position, three's in a negative position and if three's in a negative then this is in a positive position. So I'm thinking that the signs here will be a plus, minus, minus and plus. Now so every position the number, every position within the determinant has a sign associated with it. Now the way I might expand the determinant this time, it's going to give me the same answer I would get in any other way, in the other way I expanded it, is I would take the number five and the number five is in a positive position so I'm going to put a plus in front of that and if I remove the row and the column that the five is in, I'm going to multiply it by the number four. Five times four. Okay now I come to the number two and the number two is in a negative position so I'm going to put a negative in front of it so that'll be a subtraction sign and if I remove the row and the column that the two is in, I'm left with a three, I'm going to multiply it by three. So this gives me five times four minus two times three. Now you know this time the reason I have a minus here is because I say two is in a negative position and this gives me twenty, take away six, twenty, take away six is fourteen. You notice that's the same thing as taking the product five times four minus the product two times three. Now the number four is said to be the minor, M-I-O-R-N, M-I-N-O-R, is said to be the minor of the number five and the number three is said to be the minor of the number two. That is when I remove the row and the column the two is in, I'm left with this minor right here. Now when I include that sign either the plus or minus sign with the minor then this is said to be the cofactor of two and four is said to be the cofactor of five. Let's go to a graphic right here that will summarize some of this information. Okay so you notice here for the two by two case I've indicated that the signs are positive and negative alternating on the first row. They also alternate going down plus and then minus but along the main diagonal they're all pluses along there. And so when I go to expand a two by two determinant this is the formula I think you want to use right here that's multiplying a one one times a two two that would be this product along here minus a one two times a two one. Now before I talk about the three by three case let's work an example. So let's go to the green board. Suppose I have this determinant four one negative two zero three five minus one four minus minus one again. Okay I'd like to expand this. Now this time I'm thinking that the number four on the first row first column is in a positive position and the signs alternate going across minus and the negative two is actually in a positive position even though it's a negative number. And then going down the signs alternate plus minus plus and the signs continue to alternate going across and going down. So if I were to expand this along the top row I would write down four times well let's see now four. Four is in a positive position so I'm going to emphasize that by putting a plus in front of it and now if I remove the row and the column that the four is in I'm left with this two by two determinant right here that's what I'm going to multiply the positive four by three five four negative one. Now when I go to the number one one is in a negative position so I'll put a minus in front of that I'll be subtracting and if I remove the row and the column the one is in I'm left with the two by two determinant zero five negative one negative one. And then finally when I go to the negative two negative two is in a positive position so I'll put a plus in front of that and I'll multiply by let's see the two by two determinant that remains when I remove the row and the column that the negative two is that the negative two lies in that leaves me with zero three negative one four zero three negative one four.  So what we have here is four times negative twenty negative twenty three minus five minus six so this is going to be minus ninety two minus eleven is minus one hundred and three minus one hundred and three. Okay so for the case of a three by three determinant we take the very first entry a one one times its co-factor minus a one two times this two by two determinant plus a one three times the two by two determinant that remains. Let's go to another example suppose we want to evaluate this determinant let's say we put in one two negative four zero five negative one two three two. Okay so to evaluate this I'm going to expand along the first row and I'm going to take one times its co-factor its co-factor is the determinant five negative one three two five negative one three two. Okay now I go to the two the two is in a negative position so I'll go ahead and just put a negative out in front and that negative is actually part of the co-factor and the co-factor is zero negative one two two zero negative one two two and then I go to the four so now I add on a negative four well instead of adding on a negative four I think I'll just write minus four times its co-factor now there won't be any sign change here because negative four is in a positive position so I'll just write zero five two three zero five two three. Now to continue this I have to evaluate each two by two determinant and I do that by using the shortcut rule that we just saw a moment ago I take this product minus that product one times ten minus negative three ten minus negative three minus two times zero minus negative two zero minus negative two and then minus four times zero minus ten. Well now we just have simple arithmetic here it's going to be one times ten plus three is thirteen minus two times that's going to be two minus four times negative ten so this is thirteen minus four plus forty let's see that's going to be fifty three take away four is forty nine is forty nine is the value for that you know one of the reasons that these these values are referred to as determinants is because this answer is determined once you fill all those numbers in in other words once those numbers have been those nine numbers have been established then this answer has been established even if I haven't computed yet it can't change as a matter of fact if I expand on any other row or any other column I'll get that same answer let's just try doing that see we got forty nine here let's keep that in mind I'm going to put a forty nine up here beside it and let's expand along another row or another column and see if we if we get the same result so I'll just rewrite this determinant one two negative four zero five that's a five there a little hard to read and negative one and two three two suppose we were to expand along the middle row along the middle row let me just circle that one so we see it highlighted now my first entry is zero by the way is zero in a positive or negative position negative it's in a negative position so when I write down this determinant the cofactor determinant two negative four three two I'll put a negative out in front to adjust the sign of course that product's going to be zero anyway so there's no real real effect on that next I go to the five I'll put a plus five and I'm putting a plus because fives in a positive position and it's cofactor if I delete the row in the column will be one negative four two two and finally I come to negative one now let's see negative one is in a negative position so I'll be have I'll be changing the sign of the cofactor what I'll do instead is just change the sign of the negative one make that plus one and it's cofactor will be one two two three okay well without even evaluating this determinant I can see that that product's going to be zero so let's move to the next one plus five times the value of that determinant is going to be two minus negative eight and then the next one is one times three minus four so we have five times ten plus one times negative one just negative one so that'll be fifty minus one is forty nine again yeah we got the same result again I would have gotten that same answer if I'd expanded on the bottom on the bottom row and you know you'll get the same answer if you expand along a column as well as a row let's just try doing a column expansion and verify that for one of the columns so I'm going to take out that mark let's say we expand along the last column over here I'll just use that one so I'll start off by writing down a negative four well I want to change the sign of negative four leave it alone in other words will I be putting a plus or a minus in front leave it along yeah it's a positive position so there's no sign change here and it's cofactor is zero five two three okay then I come to the negative one now the negative one is in a negative position so I'll make that a plus one and it's cofactor is one two two three and then we come to the two let's see now the two is in a is in a positive position and so we'll leave it alone plus two times its cofactor which is one two zero five okay now evaluating the two by twos we have negative four times zero minus ten zero minus ten plus one times three minus four three minus four plus two times five minus zero okay we'll keep our fingers crossed here we think the answer is forty nine let's see if it works out negative four times negative ten is forty and then one times negative one is negative one and then two times five is ten yeah there's a fifty minus one is forty nine once again you notice we get the same answer but we're not adding up exactly the same numbers to get it so as you pick a different row or different column these products and sums individually turn out to be different but the result is the same so that's one of the reasons why these are called the determinants because the answer has been determined once you put the numbers in there you know if you were going to be working this one in the future what would be a shortcut for choosing a row or a column to expand along so that you could get the so that you could come up with the answer what what shortcut would you suggest Stephen finding rows or columns that have zeros in them so yeah you can not even have to determine it because it's going to be zero at least at least some of the products you could leave out because that would make a coefficient zero you know a minute ago when I expanded along the second row here I didn't have to worry about zero times its cofactor because the product's going to be zero anyway so I think the shortest way usually is to pick the row or column that has the most zeros if you see a row or a column with say two zeros by all means go for that one because there's only one product that you'll have to actually carry out operations to make one of the another number is yours that you could yeah yeah as a matter of fact those those three elementary row operations that we used earlier to reduce augmented matrices we're going to be applying those to determinants in just a moment in fact that's that's coming up very quickly here that that's a good suggestion okay we can we can summarize this with the fact that says the value of the determinant can be found by expanding by cofactors along any row or column not merely just along the first column like we did in the in the very beginning okay now let's consider the situation of a matrix rather than a than a determinant and suppose I have a determinant or rather a matrix a and a matrix b and I multiply them together now if I take the determinant of that product that's the same thing as taking the determinant of a times the determinant of b separately and multiplying together in other words this is a number terminated a times the determinant of b gives me the determinant of the product matrix that this is a theorem in algebra that we haven't proven but I'm going to make use of and the way I'll make use of it is as follows suppose I have a matrix that has an inverse which we call a inverse that product is always an identity matrix now if I take the determinant of this product I should get the determinant of the identity matrix and you remember in the identity matrix you have ones along the main diagonal so this is going to look like a one and then a zero and a zero and then a zero and a one and a zero and etc if it were if it were a three by three it may not be a three by three but the idea is I get ones along the main diagonal now if I if I evaluate this determinant by expanding along a row or column I think you'd find out that its value is always equal to one so what we have is that the determinant of the product is always equal to one and if I use that rule that I just mentioned a moment ago the determinant of the product is the product of these two determinants separately and so what this what this tells me is that a that if a matrix has an inverse then the determinant of matrix A cannot be zero because if this had been zero there's no way I could get a one for an answer so what I can conclude from this is if matrix A has an inverse then the determinant of matrix A cannot be cannot be zero because if it were zero this product can never be one as a matter of fact you can actually get more information out of this than what we're going to use what we can actually determine is that the determinant of A inverse is equal to one over the determinant of A although I don't think we'll be using that fact in this course see what I've done is to divide by the determinant of by the by the determinant of A so the determinant of the inverse is the reciprocal of the determinant of the original matrix okay let's go to the first to the next graphic and we have a theorem that summarizes what we've just established here and that says that a square matrix A has an inverse if and only if the determinant of A is is not zero now this is another reason why determinants are named the way they are because it's with the determinant that you can determine whether a matrix has an inverse or not I've come back to the greenboard suppose we just take this suppose we take this matrix I'll think I'll call this one matrix B and let's say matrix B is 5 4 negative 3 and 2 and we ask the question is B an invertible matrix you may remember there was another word we use for this we might also ask the question which is equivalent to it is B a non-singular matrix you remember non-singular minute has an inverse so I'm asking two questions but they really ask this at the asking the same asking not the same idea well to find out if this matrix has an inverse or not I merely have to take its determinant and see if I get zero if I get zero it doesn't have an inverse if I get anything other than zero it does have an inverse so if I take the determinant of 5 4 negative 3 2 let's see that value would be 10 minus negative 12 and I get 22 now what's the most significant fact in regard to 22 at this moment it's not zero 22 isn't zero so as long as it isn't zero the answer is yes B is invertible now that means we might proceed by the methods we saw in the previous episode to calculate its inverse but but what this does is it gives us a way of deciding whether there is an inverse before we go to all the trouble of actually computing it or not okay now I'd like to show you get another way to evaluate determinants let's see what we've what we've seen is we could expand by cofactors if it's a 2 by 2 determinant there's a shortcut because you can just take the diagonal products and subtract them but now let me show you got another way that we can evaluate determinants and this is using the elementary operations that we performed on matrices earlier here's an example suppose we have the determinant 2 1 5 3 can anyone tell me the value of that determinant 1 is 1 yes because it's 6 take away 5 is 1 now look what happens if I invert 2 rows what if I switch the 2 rows and put 5 3 on top and 2 1 down below now what's the value of that determinant negative 1 is negative 1 yeah it's 5 minus 6 is negative 1 as a general rule if you have a determinant even if it's larger a 3 by 3 or larger if you switch 2 rows you will change the sign of the determinant by the way if you switch 2 columns you'll change the sign of the determinant what if I switch columns 1 and 2 I'll put 1 3 and I'll put 2 5 you notice still the product is 5 take away 6 is negative 1 so if you interchange 2 rows or interchange 2 columns you get the negative of the determinant value that you had before ok that's the first of the three elementary operations although now these are row and column operations let me take another example and let's try performing a different operation on it suppose we have 4 negative 1 3 negative 2 what's the value of that determinant at the moment negative 5 yeah it looks like it's negative 8 minus negative 3 negative 8 minus negative 3 is negative 5 ok suppose now that I double the first row if I double the first row only not the second row we have 8 negative 2 3 negative 2 this time the answer is negative 16 minus negative 6 let's write that down negative 16 minus negative 6 how much will that be negative 10 negative 10 ok so what did we do we took the original determinant I double the first row and I got double the answer by the way that would happen also if I double the second row I double the answer what do you think I'd get if I double the first row and the second row four times the answer we get four times the answer because we double it and double it again and this answer would have been negative 20 instead of negative 10 this also works if I double or triple or whatever a column let's say I multiply the first column by 5 that make it 20 and 15 going back the original determinant and but still leave the negative 1 and the negative 2 then this answer will be negative 40 minus negative 15 that's a negative 40 plus 15 is negative 25 and you see this is five times the original answer so if you take a multiple of a row or a column you will get the same multiple of the final answer ok that's the second of the elementary row and column operations ok now to me this is this next one is the most surprising one what if I take a multiple of one row and add it to another row or if I take a multiple of one column and add it to another column let's take an example let's take one four five three at the moment the value of this determinant is three take away 20 is negative 17 now I'm going to take a multiple of row 1 and add it to row 2 suppose I take 2 times row 1 and add that to row 2 let's see the way I usually write that I think is row 2 plus 2 times row 1 here's what the new determinant would look like I'm not changing row 1 but if I take 2 times row 1 and add it to row 2 what would the new entries be on row 2 7 and 11 7 and 11 ok and so we get here the value of this determinant is 11 take away 28 is negative 17 what's the effect on the determinant when you perform this operation doesn't doesn't look like it affects it at all and as a matter of fact that's the general rule if you take a multiple of one row and add it to another row or you take a multiple of one column and add it to another column you get the same answer now to me that's surprising because that seems like the most complicated of the three operations and yet it has no effect this is summarized in the next graphic on elementary operations you'll notice the title of this graphic is not elementary row operations but just elementary operations because we can perform these on rows or columns number one if we interchange two rows or columns in a determinant the value of the determinant changes signs yeah we saw that we went from a negative number to a positive number vice versa in our example number two if we multiply a row or column by a constant the value of the determinant is also multiplied by the constant and then number three if we add a multiple of one row or column to another row or correspondingly a column the value of the determinant remains the same now here's how I can use this to evaluate determinants using an alternative method let's come back to the green board and take this example suppose I have the determinant one one two zero three negative one two negative one four okay so I'm just picking these numbers at random and I'm thinking you know what if I were able to reduce this to a determinant that was in this form suppose I got zeros below the numbers along the main diagonal and above on and above the main diagonal it doesn't really matter but what if I got zeros below then if I were going to evaluate this determinant what I would do is I would take this entry right here and multiply it by its co-factor by the way this entry is in a positive position so its co-factor is just these four numbers so it'd be this entry times the value of this determinant up here I'll just put a box around it now the value of this determinant is just this product minus zero so it would be this number times this product minus zero or to sum it all up it's just the product of the numbers along the main diagonal so if I can make this determinant into this form where I have zeros below the main diagonal then the value of the determinant is just the product along the main diagonal it's this number times that determinant and that determinant is just this product right here okay so how can I make this determinant look like that well if I perform elementary row and column operations and make the appropriate the appropriate adjustments along the way using the facts that we've just seen I think we can get there so let's say I already have a zero here I'd like to get a zero there and a zero there how can I get a zero in this position well let's see I can take row three and add on negative two times row one row three plus negative two times row one okay so this is exactly what we saw with augmented matrices earlier but now we're doing this to determine it's one one two zero three negative one now here's where the change comes in this is going to be a zero this is going to be a negative three and this is going to be a zero right there okay I've got a zero a zero I'd like you to zero right here how can how can I do that add row two to row three add row two to row three okay so if I add row two to row three by the way you know when you take a multiple of one row and add it to another row you're not changing the value of the determinant so these these determinants are exactly equal one one two zero three negative one now if I add row row two to row three then I'll get zero zero negative one zero zero negative one okay so at this point I'll just take the product along the main diagonal and that's going to be one times three times negative one which is three times negative three excuse me three times negative one or negative three of the value of the determinant sometimes this is the quickest way to evaluate a determinant quicker than using cofactors to to come up with these zeros in the right positions let's take one more example where we do this and this time I want to take an example where I have where I can factor a number out or make some other change okay in this three by three determinant suppose we have zero two one five five negative ten and negative two one and four so if I can get ones along the main dot or just anything along the main diagonal but zeros below the main diagonal then I can multiply along the main diagonal and get the answer by the way you can also do this where you get zeros above the main diagonal and the answer will be the product on the main diagonal again well let's see I'd like to get a one where the zero is can anyone suggest a way to get a one there okay let me let me re let me rephrase the question give me several different ways you could get a one where the zero is what's what's one way we could do that you can change row two and row interchange row two and row one and then divide the new row one by five okay yeah so what Jeff has noticed is we have a common factor of five there so if we could get rid of the five make it a one one negative two if we interchange rows one and two I think we're on our way to getting a one in the one one position okay what's another way I could get a one there interchange column one and column three okay interchange columns one and three yeah then that would certainly put a one there okay we have to remember though of course when we interchange two rows or two columns we change the sign of the determinant so we have to account for that and David I think you mentioned one other way or no okay someone I thought mentioned that if you took negative one half of row three and added it to row one then you'd get a one right there okay so those would all work let's try interchanging two columns because we have the one already okay so I'm going to put column number three in the first position and I'll leave column two alone and of course we have to change column three but you know these two determinants are no longer equal because when you switch two columns you have the opposite sign so I'm going to put a negative in front which is going to correct the sign and bring it back to bring these back to being equal okay by the way what's the purpose of getting the one what why is my interest in getting a one there because it's easier to multiply by one well see the one is a lever and I can use the one to get zeros below whereas if this had been a three for example it would be kind of awkward to use a three to get a zero with the negative ten but you know let's do something differently here also instead of getting zeros below the diagonal let's get zeros above the diagonal one for one reason I already have a zero up there so that means I only have two positions to worry about I'd like to get a zero in this position and to get a zero there why don't we multiply column two column one by negative two and added to column two in other words this is going to be c2 plus negative two times c1 we've never written it that way before because we've never done a column operation quite like that before by the way when I take a multiple of column one and added to column two will it change the sign of the determinant no that has no effect but I don't want to forget to this negative I have to keep this negative up with me all along so I have to keep recording that negative from the column change okay column number one stays the same column number two becomes a zero twenty five twenty five and negative seven negative seven I need a little bit more room I think on that one but let me switch markers here and column number three is okay zero five and negative two now you notice in this second row I have a common factor so what I'd like to do is factor out that five and call this a negative five out in front so what I'm doing is I'm factoring out a common factor on that row this is using the second elementary operation and this becomes one zero zero negative two five one four negative seven negative two so using the second elementary operation row operation I'm now factoring out a common factor in bringing out to make these numbers smaller okay now I'd like to get a I'd like to get a zero where that one is what's one way I could get a zero there you could switch row two and row three yeah I could put a five there and a one in the middle and it'd be easier to let the five the one make the five into a zero than making the five convert the one into the zero so why don't we convert these two rows switch those two rows which is going to put yet another negative out in front so now we have a plus five and the first column remains the same but the third column moves into the second position and the second column goes into the third position and Stephen's idea was that you're getting a one in the middle I can use the one to create a zero there so let's now do that I'm going to take a negative five times column two and add it to column three so let's see that's column three plus negative five times column two I'll just write that right above column two so I have one negative two four oh I left off my five that's it's very easy to drop off those coefficients if you're not paying attention there and I almost did the same thing the second column becomes zero one negative two and the third column becomes a zero zero and what's the last entry going to be three it's going to be three yeah okay I have now made this into a matrix that has zeros above the main diagonal so if I take five times the product of the main diagonal that'll be five times one times one times three this answer is 15 now you know you might say well Dennis I don't know if this is really a very practical way to evaluate a three by three determinant this one turned out to be kind of long it could be that it would have taken some other steps it could have been a little bit shorter but actually where I want to make use of this sort of a procedure is in the next problem when we talk about Kramer's rule so let's move to to a whole new problem but I'm still going to use this idea or these elementary operations anyway to consider Kramer's rule now first of all let me tell you what the rule says and then afterwards we'll go back and establish what it's about suppose I have a system of equations of the form a x plus b y equals a constant we'll say the constant is r and then another equation c x plus d y equals a constant s this is a two by two system of linear equations now let's assume that this system for the moment has a solution though the word we use for that is we say suppose it's consistent meaning that there is a solution for this problem in exactly one solution then I can find the solution for x and for y by computing three determinants here's what I do first of all I compute a determinant that I'll just call capital D and it's the coefficient determinant that is it's the determinant made up of the coefficients a b c and d so I have a b c and d that's easy to evaluate it would be the product a d minus the product b c now there's another determinant I'm going to make up here I'm going to call d sub x and what the difference between this one and d is that I'm going to replace the numbers in the x column with the constants over here r and s I'm going to put those in place of a and c r s b and d and that would be of course the product r d minus the product b s and then there's a determinant d sub y it's a two by two determinant and if I go back to the original determinant I'm going to replace the y column with r and s that's why I call it d sub y so this will be a and c in the first column r and s in the second column now you might say well what's the what's the purpose of these determinants well I can calculate x by taking the ratio of d sub x over d and I can calculate y by taking the ratio of d sub y over d and this idea is referred to as Kramer's rule is Kramer's rule so if you set up your determinants d d sub x d sub y and then take these ratios you will get the values for x and y now the problem has to have a solution it has to be consistent if if this problem has no solution or if it has infinitely many solutions what happens is the determinant d turns out to be zero and you can't divide by zero so Kramer's rule will keep you from coming up with a particular solution because there is no particular solution in that case there there might be no solution or infinitely many so we say we want it to be consistent so that this determinant will not be will not be zero right here okay and by the way this works for larger systems of equations to if this had been a three by three system of equations I would have had three by three determinants and there would be an extra one called d sub z down here I think but let's work an example using a two by two and then as time permits at the end we'll go back and see why Kramer's rule works okay suppose we have the system of equations 3x minus y equals 12 and 2x plus 3y equals 19 and of course we want to solve for x and for y okay and the way I get x is I take d sub x and I divide by d so I'll need to find both d and d sub x and the way I get y is to take d sub y and divide by d okay so I'll need to figure out what is determinant d and then afterwards we'll get d sub x and d sub y so determinant d is that 3 negative one two and three it's the coefficient determinant and this gives me 9 plus two is 11 d sub x let's see can you tell me what would be the entries in d sub x negative one three 12 and 19 no let's see not for d sub x 12 yeah negative one 19 and 3 exactly yeah see what we're doing here Jeff is we're putting these constants in the first column where the x's were so the 12 and the 19 go down here and then we leave the y column alone let's go ahead and evaluate this one this is going to be 36 and then minus the negative 19 that's plus 19 and 36 and 19 is 55 55 okay now d sub y okay Jeff we'll give you another chance here what's going to be the determinant d sub y 12 here no doesn't it go okay negative let's see what we're going to do is we're going to put the 12 and the 19 in the y column okay and I'll leave the 3 and 2 3 12 3 and 12 2 and 19 and then 2 and 19 okay exactly okay so we've we've replaced the constants in the y column whereas for d sub x we put them in the x column so this gives me 3 times 19 what's 3 times 19 anyone 57 57 right minus okay here's another one what's 2 times 12 24 24 of course yeah okay so we get 33 okay now if I substitute these numbers in over here d sub x is 55 over 11 or 5 and y is 33 over 11 or 3 so our answer already is an ordered pair is 5 3 that's the solution to our problem let's just go back and check those answers up here I'll just put a 5 above x and a 3 above y substituting those and let's see 15 take away 3 is 12 and 10 plus 9 is 19 yeah that that is a solution in fact that's the only solution okay now I tell you what let me just talk about a 3 by 3 but I'm not going to work out all the determinants because that'll that'll take more time than we have but suppose I had a 3 by 3 system that look like this suppose we had x plus y plus z equals 4 and suppose we had x minus y plus 2z is suppose that's also 4 and suppose we had 2 y minus z equals 0 okay so we have a coefficient of x there that's 0 well this time I'd have to set up a determinant d a determinant d sub x a determinant d sub y and a determinant d sub z let's see now determinant d is just the coefficient determinant 1 1 1 1 negative 1 2 0 2 negative 1 and I'd evaluate that but I don't think I'll do that at the at this moment because it would take more time to do all four of these than we have now for d sub x I would take out the x column and I would put 4 4 0 in that column so this is the column that's going to change right here so I'll just put the 4 4 and 0 there the other columns are left intact okay now for d sub y I'm going to change the middle column so I'll put the 4 4 0 there and the first column stays the same and the last column stays the same and finally for d sub z will insert 4 4 0 in the z column and then this is still 1 1 0 that hasn't changed and 1 negative 1 2 okay then to get my answers to get x I would take d sub x and divide it by d and to get y I'd take d sub y and divide it by d and to get z of course I'll take d sub z and divide it by did by z but by d rather and that would reduce to give me my final answer okay now you know I think that's why don't we go to the next graphic and we can we can see that the two by two case outlined that was the that was the model that I had written on the board a little bit earlier for the two by two for a consistent linear system x is equal to d sub x over d and those are the determinants that should evaluate and y is equal to d sub y over d okay so let's spend our last few minutes here just looking at a justification for this you know we have if we have the system of equations a x plus b y equals r and c x plus d y equals s then the coefficient determinant what we call d is just a b c and d now suppose I were to multiply this determinant by x okay so I've x times this determined this coefficient determinant now you know when you multiply by x then you can multiply any row or any column by x so I'm going to multiply the first column by x because that's where you'd expect to see the x as I suppose so this is a x c x b and d because for determinants when you multiply by a constant you can multiply any row or any column by the constant okay now I'm going to take a multiple of column two and add it to column one so the multiple I'm going to take is y times column two and add it to column one so if I take y times column two and add it to column one this will be a x plus b y and this will be c x plus d y but then we haven't really changed column two so that remains b and c so all of this is column one and then this little bit is column two okay now let's see we have a x plus b y given my original system a x plus b y was equal to r so let's replace this with r and c x plus d y that's really s so let's just put an s there so what we have here is r b s d and I think I'll just bring this down x times this determinant a b c d is equal to the determinant over here I've in other words I've actually begun with this and I it's evolved into this other expression now you know what what this is is actually the determinant we call d we have x times determinant d and over here this is what we were calling determinant d sub x because you see I have the constants in the x column so x is equal to d sub x divided by d that's the justification for the rule Kramer's rule for x you know if I wanted to derive the rule for y equals d sub y over d how do you think this would this would have begun differently multiply by a y yeah and what do you think the next step would be how would I bring the y into the problem times it by the second column right instead of multiplying y in the first column multiply y into the second column and this remains a n c this becomes b y and d y then I'd take a multiple of a n c and I'd add it to the second column so I would begin to build a more complex expression in the second column rather than the first column then I would replace each of those expressions with r and s right here instead of here and we would have y times determinant d and this would be determinant d sub y now the only the only problem a person might only question a person might bring up I suppose at this point is Dennis can you divide by d I mean what if you're divided by zero well I don't think we've justified this but it wouldn't be that difficult to do so is that if this system of equations is consistent the coefficient determinant will not be zero but that's never been established but that is the case so if this is consistent this determinant will be zero I could divide by it and that's how we isolate the x and that's how we isolate the y for for Kramer's rule well you know if we sort of recap what we've done today we started off introducing determinants for the simplest case two by two and we said that you could just take the cross product the two diagonals and subtract those products to evaluate that determinant for three by three determinants and larger you expand by co-factors and in that case you pick any row or any column and you expand along it and you take each entry times the corresponding co-factor which is a determinant with the appropriate sign placed in front of it either a plus or a minus you know we didn't do a four by four or a five by five determinant but they're done in exactly the same way you expand along a row or column and let's say if it's a four by four the co-factors will be three by threes and then each of those co-factors have to be evaluated separately as three by three determinants then we saw that there was there was an alternative I don't know if you'd call it a shortcut and that's where you make the determinant into a diagonal and multiply along the main diagonal I'll see you next time in episode 26


The study found that countries with a single national authority responsible for point of care testing, PUCT, have a higher rate of adoption than those without. This is due to the fact that a single authority can manage the task of PUCT rollout and reduce the workload for general practitioners, GPs, who are often responsible for setting up quality control, training and support. This article was authored by Dionne Lingervelder, Hendrik Hofgeberg, John D. Emery and others.


Russia took action for a coup d'etat in Armenia. Russia's Federal Security Service is directly involved in destabilizing Armenia with the aim of toppling the country's government. This was reported by the Guild Hall news agency, citing a source in one of the European intelligence services. According to the source, several FSB units are involved in the effort, as well as a separate group that was deployed in Yerevan to coordinate the operation on the ground. The team is headed by Roman Romachev, an FSB operative who is an expert on the so-called active measures and operates under the cover of a front company, the R-Techno-Private Intelligence Agency. Among other things, the group is tasked with organized and coordinating anti-government protests, demanding the resignation of Nikol Pashinyan's government, providing information support for protests across social media platforms, as well as setting up provocations with the aim of radicalizing protesters. As reported earlier, after Azerbaijan ran a blitz military operation in Karabakh, a wave of protests and riots blasted across Armenia with demands for Pashinyan's government to resign. In turn, Prime Minister Nikol Pashinyan made a statement that the son of former President Robert Kocharyan, Levon, who is believed to be a close friend of Vladimir Putin, stood behind the mass protests. As reported by Armenia Today, the head of the Armenian government ordered members of his team to notify international partners that Russian intelligence agencies were plotting a coup in his country. Also, after reports emerged of Russian intelligence setting up grounds for a coup, Nikol Pashinyan dismissed the chairman of the presidential guard Sarkis Hovannisyan just four months into his appointment.


committee the whole meeting for May 16th our first one of the year. We'll do a quick roll call. Sure go ahead. I'll have to click through that. Alderperson Ackley. Alderperson Decker. Here. Alderperson Feldy. Alderperson Felicky Panesky. Alderperson Heidemann. Alderperson Mitchell. Alderperson Pirella. Alderperson Rust. Alderperson Salazar. Alderperson Rami. There are 10 present. Thank you. We'll start out with the Pledge of Allegiance. I pledge allegiance to the flag of the United States of America and to the Republic for which it stands, one nation under God, indivisible, with liberty and justice for all. All righty. We'll start out with approval of minutes from our October 18th, 2021 meeting. Do we have a motion? Sure. Looking for a motion. Motion's been made and seconded. Any discussion on those minutes? All in favor? Aye. Any opposed? Chair votes aye. Those are approved. Okay. We're gonna start right out with the... Excuse me, do we have any public form? No. Okay. We're gonna start right out with the presentation. Chad's gonna start start things out here. So go ahead, Chad. Thank you, Chair. So we're here this evening to hear a presentation from Baker Tilly, who has been retained by the city of Sheboygan to work on the strategic plan updates. So a number of you on the council that were with us last year in the last council session were part of some interviews through the different processes. There's been some focus groups, a community survey. So we felt as staff it was a good time to bring the council up to speed on where the strategic plan processes before we dive into working as staff on action steps and priorities for the future. So Baker Tilly, there's two gentlemen, Chris and David, who are doing this remote versus traveling here. They will be here in person for the final presentation in the coming. Months, but they'll be doing this presentation remote and giving the council and the committee of the whole an update as to where they are with the process. So I think with that, I'll turn it over to David or Chris. I'll be starting, Chad. Thank you. This is David. Good evening. Members of the Board of Aldermen, my name is David Eisenlauer. I'm a managing director with Baker Tilly and the overall director of the strategic planning project and been very actively involved in what you're about to see and learn about tonight. I'll let my colleague, Chris, introduce himself as well. Thanks, David. And hi, everyone. Good evening. It's good to see you again. And yeah, thank you for your time. Looking forward to giving you this update tonight. So if we can go to the next slide, Chris, we've planned an overview of the process that we've been through where we are in that process and kind of a high level summary of some of the things that we've learned over the course of this project. So on the left side of the slide, you'll see what tasks, major tasks within our work plan have been completed. We've done the internal stakeholder engagement. That's the interviews with many of you, with staff members and so forth. We've done an outreach process with several community groups, and you're going to learn more about all of these, the results of all of these in a moment. We did what we've called a resident input questionnaire online to gather information and input into the strategic plan from kind of all commerce, anyone who wanted to participate. We then did a visioning workshop session with the mayor and the administrator and the other senior members of the team, the department directors, and have got some preliminary goals, objectives that emerged from that were visioning session that we're going to share with you. As Chad mentioned or certainly alluded to, this is indeed a work in progress. So from the high-level framework you're going to see this evening, we are now working together with the staff to build out the structure of the plan and the details of the plan in terms of goals, objectives, initiatives, and priorities. And we will be then preparing a preliminary draft once that work is completed to share with you. And then based on that review and comment on that draft, we'll then, of course, create the final document and deliver that to you. The next slide is really where Chris is going to take over. He was an active participant in the outreach and engagement piece, and he'll walk you through what we learned from those various groups and individuals that we interacted with during that phase of the engagement. So go ahead, Chris. Yeah, thank you, David. And if you remember back in December, we met with, we spent about a week at City Hall and met with quite a few people. So here's just kind of a full list of all of those that we met during the discovery phase interviews and all the elected officials and the department heads with the staff and had some focus groups as well with some kind of supervisory roles as well. And we enjoyed the conversations. We learned a lot about the city, kind of the challenges, the strengths. And kind of as a result, we really observed some, some common themes. And we had some reoccurring topics come up during those conversations. And here's just a quick summary. It's about six items. You know, municipal infrastructure came up quite a bit, mostly related to the need for road improvements, concerns around a cost to make to the cost for infrastructure improvements. And some other various topic areas, but those were kind of the big ones. Another one probably that came up just about as much was the, oh, can everyone see my screen? Okay. Yep. Oh, great. Okay. Was community, excuse me, communication and engagement? And again, mostly related to the need for improvements to the website, but also kind of making sure that there's a plan in place to make sure everyone's on the same page in terms of whatever kind of communication is coming out of the city. Another big one, which I believe we're talking about a little bit earlier this evening was housing choice. We heard a lot about the efforts that the city's been making and very positive strong efforts. And we also talked about the concerns. Still about the availability of affordable housing, housing for employees, senior housing, and some challenges related to community opposition to increased housing density. Economic development and tourism came up a lot. This is a very positive thing. A lot of people know that about Sheboygan. And it was the conversation related mostly about how to capitalize on opportunities that Sheboygan greatly positioned in and ways to work better as a group with all of the various economic development and tourism groups in the region. Operational excellence is one kind of term that we've coined, but it covered a lot of topics. And it's really related to internal operations, whether it be internal organizational culture, where the city can make improvements in that area, technology improvements to make things more efficient. So a lot of kind of internal things that the city, a lot of stuff is going well, and a lot of areas were opportunity for improvement. And then the other the last major theme was quality of life. And this kind of happened, this conversation came up, mainly when like economic development or tourism conversation was happening. And it's a very positive piece of the city that there's a lot of great parks, a lot of recreational offerings, the library is great arts and culture available in the city. But it's just a lot of how to make it better how to continue to improve. So after we had met with everybody internally, we then developed the plan to set up community focus groups. And here's kind of a quick summary of all of those sessions. Over about a month and a half span or so, we hosted nine virtual group sessions to listen and learn to those in the community. And here's a list of those that we met with from small business owners, large business owners, neighborhood groups, social service organizations, tourism groups, those involved in the effort, the housing efforts, families, representatives from the school, community arts and culture groups and elevating diverse voices. And similar to the internal stakeholder piece, we heard a lot of great things and a lot of reoccurring topics came up. And here's kind of a again, just a very quick summary of those topic areas. And as you'll see shopping restaurants in the downtown region, you know, where you can increase some available different stores that are available, increase the types of restaurants that are available and how can you make the downtown not better community. And this was looking at my notes here. Community was related to the location, the strength, how that is a strength to Sheboygan and again, how to kind of capitalize on the position that Sheboygan is in. You know, it was noted that the city is making efforts to, excuse me, that's a different section. Oh, pardon me. So yes, that's kind of where the most of the conversation was related to communication, collaboration and clarification of roles and responsibilities. Similar to how it was mentioned during the internal stakeholder piece of how to best get on the same page, make sure all types of communication, everything that is being communicated, everyone's on the same page. This was related to, you know, what organization within the community is responsible for a certain, a certain area and working together and making sure that everyone knows what roles and responsibilities they have and it's clear. Business and workforce. Again, I kind of mentioned it earlier. The pardon me, lost track of my notes here. This was related to just the challenges that businesses are having to attract workers and sometimes it was related to housing, which was also noted. The talent and time to ability to attract talent. Community amenities was another big one. I mentioned it earlier, the very same themes, how supporting is greatly positioned with the great parks with the lakefront. And lastly, again, I mentioned housing and that one came up quite a bit. And then in addition to that, at the same time, as we were having these community focus groups, we sent out a resident input questionnaire. And we had some great response rate, just over 1800 responses, which is very good. We were happy to see that about of those that responded about 38% had been living in Sheboygan from zero years to 10 years, and about 61% of the respondents had been living in Sheboygan between 11 and 25 years. We were most excited of the wide range of age groups that responded to the questionnaires as you can see here. It's pretty consistent in across the board from those in their teens, mid 20s, people in their 30s and then an older. It was consistent across the board. And on the right there, you see is kind of a fun way to display the word cloud, a fun way to display the words that people use to describe the opportunities that Sheboygan, in their opinion, Sheboygan has available. And just a few to mention, we're kind of again, the downtown business is how to improve that, capitalizing on just the position, the geographical location that Sheboygan is in and the lakefront available, the need for new or affordable housing, diversity, and sense of community. So that's a very quick summary. But it was a long process and we learned quite a bit. And I'll turn it back over to David and all that we learned there is what kind of helped us drive the conversation during the visioning workshop. David, I think you might be on mute. Thank you, sir. I was go ahead and advance the slide if you would, Chris. So we gathered at the the marina with the mayor, the administrator and the senior members of the of the city staff, the director level folks to think a lot about the future vision and direction of the city and what that meant in terms of setting a set of ultimately goals, objectives and initiatives to move the city in that direction. And the first thing that we did was a little visioning activity, which involved a set of about 250 photographic images that we asked the participants to browse to look at and to to pick one that that said something to them about what their personal view and vision for the city of Sheboygan is. And you can see what some of those images look like. And then they described why they pick those images. And you've got a sampling of some of the words that they use. And I want to read them all, but they were about recreation and industry both a balance of those unity teamwork adventure, all sorts of really wonderful ideas and thoughts about about the community and how special it is.  So we then did a word cloud exercise using some online polling. So this word cloud was built by the group in real time. And the larger the word, the more it was mentioned and more times it was mentioned by the by the participants. And therefore are some of the stronger themes that came out of that visioning process. And that you can see it in this diversity and safety and fun and comfortable and growing is descriptors that the team came up with each individually. But again, where you see those larger print words, those those were really consistent that meant more than one. And sometimes many more than one individual cited those as a part of their own personal vision for the community. Next slide. We did kind of a classic exercise of strategic planning called a SWAT analysis, strengths, weaknesses, opportunities and threats. And again, this is a little bit of an eye chart here. And so I don't want to read it to you. But this is really a while we did the visioning to sort of focus on where we want to take the community. This was really a close look at the organization itself. And what what we needed? What did we do well? What do we currently do well? Where do we need to get better? What things in the external environment worry us that that that could threaten us? And we need to think about how we adapt and mitigate those threats. And then if all of those things are true, what are at least some of the of the opportunities that await us. And so those you see in the upper of a right quadrant in the little starburst things and these I'll read back to you if I can read see them. So internal operational in improvement, we can be a better organization. We can do a better job in in our investments in our capital that relates to the infrastructure issue. We need to do a better job and have real opportunities to build strong partnerships with the community. Some of those organizational relationships that Chris alluded to in the community feedback, but generally a theme of we've got an opportunity to do a better job in the partnership area. There was a recognition that you the board of Alderman represented an opportunity in terms of the kind of shift that's gone on that's happened with the makeup of the board with a young and newer council member bringing fresh ideas and how do we as a as an organization together take advantage of that. There's an opportunity to attract new business. There's opportunities in downtown. There's opportunities to do a much better job of telling our own story running rather than letting others tell it for us. We have an opportunity to build our workforce to train and development that workforce and we can continue to grow in a strategic way so that was that was a really important part of what we talked about as thinking about what we're good at what we're not so good at what scares us is that is that then what do we see as as some of the of the real important opportunities for us to grab. Next slide. So we did and you're going to see some photographs in a minute of what this exercise looked like. But it was a some of you have probably been through these in your in your work or school but a sticky note exercise and we we use the big graphical template and ask all of the members working there together to reflect on the vision and reflect on the SWAT analysis and then using little sticky notes right down sort of very specific needs. That's the black text on on here that they felt like looking out into the future the city needed to take and there you can see there are all different kinds of viewpoints and perspectives and level of detail. But that was the idea we were really focusing on what quantity of ideas not necessarily caught quality in other words we advise everyone to say look we've got we've got to enter this process and behave with an open mind and so no one criticized the these ideas. They they all got a chance to talk about them and tell people why they thought they were important and then I had them literally re shuffle all of the sticky notes in the in the clusters that represented some common themes or or or potentially goals and that's what you see in the color text. So one was to practice intentional communication and engagement and you're going to see I hope the linkage between what we heard from the internal and the external focus group. We got to get better and being really intentional about how we engage the community and how we communicate and tell our story. We got to build community alliances. That's the partnership idea. That's an important strategic goal for us. We've got to do a better job of of on the community leadership process the governance process in in interestingly on thinking about how you use your authority to appoint people to committees boards of commissions and outreach recruitment of folks to get involved in in in the city and and also to do a better job. The staff is telling itself we need to do a better job of helping the council understand what it is we do the details of the of the operations of the city. You can see advanced human capital was all about the city's workforce. You know how can we make sure that we can evaluate people well that we can employees to buy into our mission and vision that we can our leaders are held accountable and so forth. A fourth one was continuous improvement. That's what we called earlier on Chris alluded to his operational excellence. How do we operate more efficiently as a city. Infrastructure investments. A big theme we heard in the in the outreach is an important goal and then inclusive and supportive culture. So that's about the organizational culture and in terms of making sure that we're engaging and and being inclusive with our with our employees. Building a robust the I plan working to eliminate any negativity in our culture over time. So those were the big ideas that came out of this next next slide. Chris. So we started looking at that list and this was getting fairly close to the end of the day and you say you know what most not all but most of what we talked about during our full day together. What we could really look at as as being as much about the organizational needs and less about the community priorities. And so we we we started to talk about that and concluded that the strategic plan as we develop it really needs to to to be written in such a way that we acknowledge the the importance of both of those and say and so we we invented some new terms and these are still that's still work in progress. But we said that what what the sticky note exercise told us was that our priorities as an organization around stewardship of our resources around nurturing and growing and strengthening our human capital. It's about in building the kind of culture that we want to have an inclusive diverse culture within the organization and this continuous improvement idea that always getting better and always getting will be more efficient on the community side. They said here the kinds of things that we believe that we were hearing from that outreach and we also went back and took a look at the 2018 version of the of the strategic plan to see what kind of persistent ideas that are relevant then might still be persistent today. And again we came up with 4 broad categories. The community wants us to elevate the quality of life and should work in and that has to do with revitalize neighborhoods that's where housing fits. We need to to leverage and improve and expand our parks and open space an important part of our community identity and potentially also an economic development. As yet we talked about the development of 3rd spaces this had to do with new kinds of venues dining entertainment those kinds of things and then strengthening our parks and leisure services recreation kinds of things all of those being sort of a sub subset of what is quality of life. You said we want to we the community expects us to have a robust infrastructure both in in terms of the of traditional infrastructure like streets and utilities but also facilities. We want to think about our growth to make sure that we're growing in a sustainable strategic way and then again the community expects us to to to build a build a build a sense of belonging among among the people and the businesses of the community and building and strengthening alliances. So this this is what we end of the day with as as these really kind of 8 priorities half of which are focused on the organization and half of which on the community. So next Chris. So well, let me let me back up for a second. So what next after after those goals and objectives and we alluded to this earlier early on when we now have to build out sort of the under structure of that what specific things in the way of supporting objectives and action items need to be taken in order to drive that that vision and to accomplish those big picture goals. The next few slides that would just page through kind of quickly or just to give you sort of a sense of a feel of what this workshop look like we did it. I think as I mentioned it to the at the Marina. There's a part of there's Chris leading the the SWAT analysis and our other colleague who is with us. This is where we were we're really talking and there was some kind of the spirit of dialogue about about what we're good at and maybe not what we're good at and and having people sort of clarify that you can see here that we had lots of different ideas about that and we ultimately sort of shifted shifted these things around to make sure that we had them right that the strengths were right and the weaknesses were right and so forth and Chris apparently enjoyed doing it. I've got a really good feeling at the end of the day is all happy. So that's an overview of where we are and Chris will kind of wrap us up here. Go ahead Chris. Oh no, I don't have anything else to add that was we've enjoyed working with you guys up to this point and continue to look forward to kind of defining everything and fine tuning this. We're happy to answer questions any any feedback or questions we want you as the policymakers to really have a sense of where we're going directionally and to make sure that we that we're not missing things that that you consider really important before we spend additional time sort of building out that framework. If if the framework in in in pretty close to right. So we're looking for your feedback at this point. All righty. Well, we'll start taking questions. I see Barb's got on the queue right away. So I'll put her right up there. Thank you so much. This was very interesting and of course it piqued my interest on having a hard copy of this. Are we going to get one of those so that we can look it over and you know maybe do some strategic planning for it? Is that directed to us or to the to the staff on the distribution? Chet's got his handle. So we're going to let him let him answer that. Thank you. So we can forward this presentation digitally. And if you need a copy printed, I'd be happy to print a copy of it after this presentation, whatever, whatever would help with you guys analyzing kind of what's here so far. OK, I'm OK with the digital than it's on my computer. I could look at it whenever I want. OK, next up is Bertha. Thank you. I was intrigued with the fact that staff had the ability that they were interviewed and then they had the ability to cohease as a group and move priorities and think about priorities. The alders had the ability to do one on one. Will the alders likewise get an opportunity to cohease as a group with sticky notes, et cetera? Because I think our perspective is different than staff perspective. I expect that we'll end up with the same with the same big chunks. So I don't think we need to reinvent chunks. But will we have that opportunity? Is that is that in the future? So what I would say to that is it's very difficult to do that. It's going to have to be strategic because it needs to be a public meeting because we have a quorum of the council present. And I think that's why we went the route of trying to work with the department heads first to try to hone in on this a little bit. I guess if there's a way of being able to do this in a public setting with a agenda posted and meeting open meeting laws, I mean, it's something we could look at. But I know that we've struggled in the past with the quorum issues. And I don't know if Chuck Adams has any further thoughts on that, not to put them on the spot. But that's kind of the challenge, I think, is the open meeting requirements. Chuck, you want to answer that? Well, it would have to be in a public meeting. But that's, I mean, anytime you meet, it's in a public meeting. So I don't necessarily see that as a big barrier. You would post the meeting and you would do things in public. That doesn't mean that you have to give the public the opportunity to speak just in the same way as you do a public forum. You can limit it. And you would just do the same thing. It's just that you might have an audience as you should. Thank you. Chris? Yes, would that be possible to bring up the presentation again? Sure. Chris, can you share that again? You have a particular slide you want us to go to? Fortunately, I didn't know down which slide that was. Okay. Topic, do you remember? Yes, so the question I have, and maybe then you will be fine, we will be able to identify the subject, is some of the statements that you took note of are slightly obscure to me, at least three or four of them. And I wonder if you gave us the extreme synthesis of what you heard and, but you have more content to those statements or those statements are just the way you receive them and you would not be able to expand on those. For example, I believe there was one slide where there were two or three statements where I just cannot relate, I don't know what they mean. And one of them refers to the negative cultures. So would you be able to add some more understanding of that statement that you got during the interview or work or is just the statement itself? Let me take a swing at that and then maybe either Chad or Todd might want to weigh in on that what you are seeing in all of those slides from the community engagement process is our attempt to distill into very few words, which was an extensive amount of input and data and there is additional detail that would flesh those concepts out. And so that was just our attempt to figure out a way to summarize the results thematically and not to put in this slide presentation all of the detail that we have, but there is significant detail behind all of that. So you would be able to expand on some of these for us to understand it better. Yes, we certainly could do that. We spent a good bit of time and I should have mentioned this when we met with the staff at the marina in going over some of that more granular detail that we had and trying to get real real clarity and they asked really, really good questions about that data. So again, it does exist. When would that be a good time to have that understanding? Okay, go ahead one. So for the purpose of this discussion, David and Chris, can you go to the slide that's got the multicolored headings and then the black underneath it with multiple things on it? That's the one where it references, I think the concerns of Alder, Perola. Yes, the bottom on this slide is where it says work to eliminate negative cultures. Is this the slide you're referring to? Yes. So I think what the, and David is answering it, so I think it's a commulation of everything that they've seen to date and heard and put into these and then these working strategic goals and objectives, correct me if I'm wrong, David, but this is really taking that information that you presented in all the engagement and working with the staff and coming up with some buckets to try to put this stuff in. Is that correct? Yes, and thank you, Chad, for recognizing that this was the particular slide that the Alder person was looking at. So the extensive amount of detail that I was talking about a minute ago was related to what we learned in that engagement process, both the internal interviews and in the focus groups and in the online questionnaire. This material that you're looking at right now came from that sticky note exercise. And so this, I do recall some of this, just how it sort of unfolded in the workshop. There were several members of the management team that in talking about what kinds of things do we need to be better at is strengthening our culture and recognizing that there might well be pockets within the city organization that could use some strengthening and some positive movement in the culture. So the detail behind that is not as voluminous as the community engagement. It's on a series of literally post-it notes that we use to summarize and characterize what those notes meant. Again, I hope that answers your question, because I answered a different question now than you asked. Go ahead, Grazia. So you did, actually, thank you. And so that slide, then, if I understand it correctly, is a combination of the internal sticky notes exercise and everything else, right? So would you remember? Yes. I'm sorry. I stepped on you. Please finish your thought. I'm sorry. Would you also, I'm just curious, know that always be true, truthful, I believe, to ourselves? I don't have the slide right now. Where does that come from? Always stay true to ourselves. It's another one which is kind of vague. And so I am trying to understand the content, the substance of it. Always stay true to ourselves. Right there. Up there at the top. I'll have to be honest with you all, the person. I don't have a particular recollection of exactly where that one came from and what it represents other than it related to culture. And I think it had to do with being public servants and being proud of being public servants and making sure that we do the things that we know to be right. At least that's how I would interpret it. And if any on the staff want to correct me on that, that'd be fine. But that's certainly how I read it. Chief Tomagowski has something to add to it. Sure. I recall the discussion being, we were talking about many different things, but some of them, what were other cities doing? What direction should we move in? And one of the thoughts that was brought up is that we have to remain true to what Cheboygan is to, and not necessarily try to change the strengths of what our community already offers. Thank you. Thank you both. Okay. Alrighty. I guess Bertha's got comments here. Thank you. That the slide that was just up was fairly meaty, and I could relate to that. But then we got to the end, and we had a slide that had organizational priorities and community priorities. And I didn't see very much of a bridge between the slide that we just saw and the organizational priorities, stewardship, human capital, inclusive culture, continuous improvement. You can open any business book and you can see those four things in it. So how did you, how did you arrive at that? Okay. So part of that, and this is still a work in progress. And so the wordsmithing isn't necessarily complete. But it was as I attempted to convey a recognition that there was something incomplete in everyone's mind on that multicolored slide. That it did not describe in a comprehensive enough way where all of our priorities are. And that just as a part of the process or any number of reasons, who knows, we've stepped back from that group of sticky notes and asked the question, is this complete? And the answer was no, it wasn't complete. The answer is, is that we talked more about the organization and our priorities as, you know, leaders within the organization, which you would expect and less about, not with, not completely absent, but less about what we need to do as an organization to address the priorities of the community. So that's when we just made the recommendation to the group that what we ought to do is in effect bifurcate those ideas and say our strategic plan will address both the needs of the organization, the strategic needs of the organization and the strategic needs of the community. And that those words that are there, we just sort of brainstorm them on the spot, just to capture the idea of we've got to address both and not just one. So there's some crossover. If you look at the multicolored slide, you'll see, for example, infrastructure is on there. I think that's one that sort of found its way to the community side rather than the organizational side. So it's not completely true that everything on that multicolored slide was only internally focused, but it was more heavily weighted. I'm sorry I talked so long, but that was the way it unfolded. Follow up on that. Thank you. And the thing that I see that dropped off was communications and engagement. That doesn't show up anywhere in organizational priorities or community priorities, but it was one of the seven or eight on those colored places. So I guess my question is, have you lost something by trying to condense it so much? That's a possibility, all the person, and we'll certainly go back and make a note of that and go back. I can tell you with real confidence because of exactly what you're noticing is that improvement of communication and engagement will be prominent within the plan. Sort of the art of doing these plans, it's more art than science, maybe it's a craft, is making sure that you find the right level for each idea. So some ideas rise to the level of a strategic goal, which is going to be something that's very durable and persistent, and at the opposite end, there are other things that are really tasked that you might finish in the course of a year or two. So we've got all of those ideas within, I'll use your great word for various buckets, and it'll all be there in the plan. But we'll take, and don't let me put words in your mouth, we'll take from your comment that you would like to see that communication and engagement rise to the level of one of those eight or nine. Like you said, there's no magic number there. Top-level goals, is that a fair recap? Well, sure, but that's, my observation was a little broader than that. I just want to have captured all the richness of all of the input, and it just felt like after all the sticky notes and everything, and you came up with organizational and community priorities that it got sanitized, and I don't want it to get sanitized. And I do understand that you need to focus so that you can actually build a strategic plan. I don't have any problem with that. So I guess maybe go back and craft some words and make them richer than a business book index. Okay. Thank you. That's exactly the kind of feedback we're looking for and we appreciate it. Okay. Anyone else have, I guess that's it. Any other comments from anyone? Okay. Then Chad, do you want to have a little bit of follow-up with this at all, Chad? Or is it pretty much? I guess, and I'm going to throw Todd in the mirror. Is it the wish of this group to spend multiple hours trying to go through an exercise similar to what the department heads went through? And I'm not sure if it's in the evening or it's a Saturday or a Sunday to try to engage in or do you want to get, you want us to, I'll tomorrow morning send you this slide deck and if there's anything else that comes to mind, you share that with staff and we share it with the consultant. So I guess the question is, you know, what is the feel of the committee of the whole from, you know, this point forward? What we had in the plan was that, you know, we would take any input that anybody has and let you have some time to digest it and give some additional input if needed, have another committee of the whole meeting if that's what it takes or, you know, or just share that with staff and then we'll, you know, take it to the next step and try to get a little bit more detail on the strategic goals and some of the priorities and next steps related to laying that out and then come back and have dialogue related to that. I guess where we're at is we didn't want to spend a ton of time putting a lot of detail into all of this stuff. If the council wasn't on board with this high level stuff. So that's why we thought it was a good point to come at this time and share what's been received to date. So I guess we need some direction from this body as to where you would like us to go from here. Okay. More than we'll send it right to Todd then and let's see his comments. Thank you, Chair. This was to bring the information together. This is the second, you know, official strategic plan. So it's very difficult. There's a lot of a lot of data. I guess my suggestion to the council would be that we, we do a committee of the whole and we can kind of get the elders to kind of participate kind of like the department heads did and maybe take that information and review it as a group to get that, that inter, that, that involvement. It's a lot of data. It's a lot of work, but we, we need input from the council on what steps you'd like us to take next. Mayor. Thank you, Chairman. And obviously, you know, only a few, few folks have spoke, but I think it's just, you know, it might be helpful to go down the line really quick, just to get a good sense of what everyone is thinking. I mean, this is the strategic plan for the next few years, not only for the city as an organization, but the city as a leader and the 50,000 individuals that we represent. So when we get a kick at the can, we got to make sure that we're kicking the can and doing it right for all the different components, communications, operations, how we're spending our money, how we're fixing the roads, how we're making our community safe parks.  So we got to do it right. So, you know, we don't, you know, I know I brought this up during the process. You know, at what point do we bring in the council? Because I get the general sense that as elected officials, you want to, you know, have a kick at the can and to say three or two cents in in the process.  Well, you're spending this money. What's this project for?   Unless you want that, I know I'm busy, but, but it could be a condensed version, you know, two hours. Committee, the whole is the platform to do this stuff.  And then you just, blah, blah, blah, blah. So I'll stop. But that's just my two cents. We want your input. We want your advice. If we're going in the right direction, what are we missing? You know, how we can fine or tune the engine and just make our community a better place. This is the opportunity to do it. So. Okay. Well, then I'm going to follow up, follow the line and Grazia is next one online. Well, I'll, I'll, I'll get you the queue. You'll be next. Okay. Gracias. Yes. I do agree that we want to be active participants as a common council as much as possible. And that is also the reason that I, I think that we may need a little bit more again. The, the word that comes to mind is expansion on some of the items that we have read tonight, learned about tonight. Because if the concept are presented in, I understand the idea of the synthesis, but if the concepts are presented in, in a vague format, it's very hard to relate to those. And it's very hard to understand what is the need behind it. So that is what I think again, I think yes, we should be, be engaged as much as possible and also be provided a little bit more content behind the, the synthetic versions of the findings, in order to relate to, to those better. Okay. Thanks, Grazia. Bertha, here's your chance. Thank you very much. And this might be for our city attorney. It would be my, my wish that we convene, you know, at, at the marina and do it like on a Saturday morning for three hours max because it's a, this is a fairly constrained space in order to do that. So is it legit that we could post a meeting and say it's a strategic planning meeting and hold it elsewhere? Meet library, wherever. As long as it's in the city, you're good. And it's public. As long as the public can get there. Thank you so much. Okay. Alder Bruce Mitchell. Thank you, chair. I just had a couple of comments on one theme that I've been hearing. The council being involved in this process is most certainly important to us where our role is setting policy and this is somewhat going to dictate the direction that policy is going for the next half decade or so. But one thing looking back to the SWAT analysis session if we had a city staff doing that and spending what looked like a good amount of time, they're here more often than we are. They're the ones that are working in the building every day. And I just want to make sure that we take into consideration that the opinion of city staff shouldn't just be set over to the side. If we do a planning session ourselves because that's giving us real insights into what the people leading and working in all the departments believe changes. I guess the general direction needs to be for them to be the most successful. So I would caution us against duplicating anything that's already been done unless we already have a plan and how we are going to integrate the outcome of the two of them. Alrighty. Well, I would agree with that. But I also think we should have, like you said, a short session, you know, a couple hours that we should go through some of the stuff, not putting aside staff following through and looking at it, you know, kind of looking at it under a microscope and just taking a quick quick look and going through it all. Does anyone else have comments? Chad, go ahead. I guess I just have a question for David and Chris, given that you guys do a lot of this community engagement across the nation. Do you have thoughts on a means of doing this maybe electronically or some other means of doing it through some type of engagement type software or something where, you know, people wouldn't have to necessarily come together but could still put their input on some of the stuff that's been shared or I guess there's people shaking their head. They don't want that. So don't worry about that. Director Meeble, there you go. Go ahead. The department heads met, you know, and if you looked at or heard tonight's presentation, a lot of the discussion focused on internal operations. The, at the end of the meeting we did agree, and I agree there needs to be that bridge between what is the external, the larger community needs and how do we fix the internal and bridge that gap. So that's why I think when you saw most of the findings said there was a lot of internal focused. The last time we did this, I really think, you know, we really should talk collaboratively, council, staff, and you talk about being more inclusive and thinking together and listening to each other. What an opportunity to, let's focus now on the community aspects. From your perspective, we could also provide input in terms of what the internal internal challenges are and what we need and what the gap is. And that way we are building at least a closer relationship in how we're going to advance the community. Thank you, David. Barb, you had a comment. Yep. I agree and that was my thought on this. I would rather go with a meeting of the elders so we can discuss, you know, where we're going ahead because we are the ones that make that decision and I'm never in favor of just a handful of people leadership making that decision without letting the other elders in on what's going on. Burt and I talk about that pretty often about, you know, well, how do we get the other older people in this so that they know what's going around. So I think I'd like to have a little get-together, first a little time to absorb all that information and then to get together and put your two cents worth in. Okay. Thanks, Barb. And I agree with that sentiment. Anyone else have any comments, Chad? So let me ask you this question, is the majority of the group by a head shake supportive of doing this on a Saturday morning or is this an evening after hours type of thing because that will help us try to hone in with travel plans of the consultant because David is in Texas. So when would you prefer to do this, what would work the best in everyone's schedule? An evening workshop or a Saturday workshop? I'm open. I'm probably, I guess I would be prefer evening, but I would be open to Saturday myself. Do we have any? What a doodle-pull. Show of hands. Who wants to do an evening? Three choices. Evening Saturday or no preference? Evening? Evening. Evening raise your hand. Right. Two, three. Saturday? Saturday. One. One. No preference. All right. Well, I guess it's the evening because there was three versus two. Yeah. So, I mean, and if worse comes to worse and we have Saturday is the only option, then we'll go with Saturday. I would just say this. If I might, we will work with you and whatever form is the most useful and productive for you. And, you know, and we will, of course, be there in person and we'll do what it takes to make that happen if that's your decision. David's excited to come visit Sheboygan again. I am. I am. Picking up surfing. Aldous Salazar has a question here. Thank you, Chair. We'll make sure to bring you some mud pies or some good bakery. Okay, just to echo some comments that I've heard that I think are really important. I appreciate what Trey said about the SWAT analysis. I think that's important for us as Alders to realize that the staff has done that. And I think that's important. I love to doing a SWAT analysis, but I also want to be thoughtful and mindful that the staff have done that. So I guess I would ask sort of Baker Tilly or Chad, whoever's sort of leading this, to really think about how the Alders can engage and how we can sort of blend both discussions together. I appreciate David saying sort of having directors there and having this conversation about sort of focusing on the public, but I also want to make sure we're mindful of some of the internal operations because that will also hit budget. And as Alders, we need to sort of think about that as well. So I know that's sort of a task at hand, but that's sort of my synopsis of sort of what we came out of today. So thank you. Thank you. Bert? You? All of the focus groups, the seven or nine or however many focus groups there were, those are our community. Those were very carefully chosen. They were representative of people in the community who were not staffed, who were not elected officials. That's the meat of what we can do because those are our constituents and they came to us in various permutations of what they do every day in their lives. So that's the part that I would like to focus on. And then we correlate that with the city's focus on, we know day-to-day operations and we've done this for years and we know you do and we appreciate what you do. It's those other constituents that we have to really grapple with. All right. Any other comments? Questions? Okay. So he knows the evening is late and there's no more questions. I guess we can leave the gentleman from Baker Tilly go and I will entertain a motion to adjourn. Motion to adjourn. Motion is made and seconded. All in favor? Aye. Any opposed? General Tye, that is adjourned. I just wanted to see just for the...