Why is (AB)^-1 = B^-1A^-1 in matrices, and not (AB)^-1 = A^-1B^-1

Answer 1

let A be an m by n matrix and B be a n by p matrix. Then A*B is defined. Also, A^-1 is an n by m matrix, and B^-1 is an p by n matrix. So B^-1 * A^-1 is also defined, but A^-1 * B^-1 may not be.

Or, in other words, the commutative property of scalar multiplication does not apply to matrices.