Question

Explain why a matrix that does not have the same number of rows and columns cannot have a multiplicative inverse.

Answer 1

If an m x n matrix A (where m doesn't equal n) had a multiplicative inverse, then


A*A^(-1) = A^(-1)*A = I


where I is the identity matrix


But multiplication isn't defined because the number of columns in A is not the same as the number of rows in A^(-1)

Answer 2

So A has to be square