I have two homework questions I am stuck on. Any help appreciate.

1. Show the sample correlation matrix R of data Xnxp is the covariance matrix S of the standardized data Z of Xnxp

And to standardize a matrix, I take each element and subtract the mean and then divide by the standard deviation. One point of confusion I have is this; am I using the mean/standard deviation of all elements of the matrix or am I suppose to use the mean vector associated with each column?

So am I just supposed to standardize the data X, find its covariance matrix, and that should equal the correlation matrix of X?

2. Show the relationship between the determinants of S and R is:
|S| = s11s22...spp|R|

One method I tried was to use det(AB) = det(A)det(B)
Or in this case, det(R) = det(R1)det(R2), where R1 = S in the hopes that I could solve for R2, and gets its determinant which would hopefully equal s11s22...spp. But I got stuck.

Thanks for any feedback/help.