The inverse matrix was only defined for squares1/ Show that if a matrix has an inverse then it must be square.

I do not know if you discussed that but the ring of matricies over the reals has no divisors of zero.2/ Let A and B be square matrices such that AB = 0. Show that A cannot be invertible unless B = 0.

Meaning that if,

AB=Z where Z is the zero matrix then,

A or B must be zero matrices.

A zero matrix never has an inverse (because anything multiplied by itself is zero matrix, just like dividing by zero).

Thus, if we want A to have an inverse it cannot be zero this causes B to be a zero matrix (disjuctive inference).