]> 32.3 The Inverse of a Matrix

32.3 The Inverse of a Matrix

If two square matrices M and A have the property that M A = I , (in infinite dimensions you also need the condition that A M = I ) then A and M are said to be inverses of one another and we write A = M 1 and M = A 1 .

A wonderful feature of row reduction as we have described it is that when you have a matrix equation A B = C , you can apply your reduction operations for the matrix A to the rows of A and C simultaneously and ignore B , and what you get will be as true as what you started with.


Thus, suppose you start with the matrix equation A A 1 = I .

If we row reduce A so it becomes the identity matrix I , then the left hand side here becomes I A 1 which is A 1 , the matrix inverse to A . The right hand side however is what you obtain if you apply the row operations necessary to reduce A to the identity, starting with the identity matrix I .

We can conclude that the inverse matrix, A 1 can be obtained by applying the row reduction operations that make A into I starting with I .


Suppose we want the inverse of the following matrix

( 1 3 5 6 )


( 1 3 5 6 ) A 1 = ( 1 0 0 1 ) ( 1 3 0 9 ) A 1 = ( 1 0 5 1 )


( 1 3 0 1 ) A 1 = ( 1 3 5 / 9 1 / 9 ) ( 1 0 0 1 ) A 1 = ( 2 / 3 1 / 3 5 / 9 1 / 9 )

and the last matrix here is the inverse, A 1 of our original matrix A .

Notice that the rows of A and the columns of A 1 have dot products either 1 or 0 with one another, and the same statement holds with rows of A 1 and columns of A . This is of course the defining property of being inverses.



If your original equations had the form M v = r , by multiplying both sides by M 1 you obtain v = I v = M 1 M v = M 1 r , so you need only multiply the inverse, M 1 of M by your right hand side, r , to obtain a solution of your equations.

If you think of what you do here to compute the inverse matrix, and realize that in the process the different columns of M 1 do not interact with one another at all, you are essentially solving the inhomogeneous equation M v = r for r given by each of the three columns of the identity matrix, and arranging the results next to each other.


What matrices have inverses?

Not every matrix has an inverse.





Thus the absolute value of the determinant of M is:





In 0 dimensions we give whatever it is the value 1.



