Answer
If A is not a square matrix,
and B is the multiplicative inverse of A,
then the products AB and BA do not result
in the same multiplicative identity matrix.
Work Step by Step
Let A be an m$\times$n matrix.
In order for B to be the multiplicative inverse of A
B must an n$\times$m matrix,
because AB and BA should equal each other, and
they should equal the same
multiplicative identity matrix, which is a square matrix
$AB$ (order: m$\times$m) and $BA $(n$\times$n) should have equal order,
that is,
$I_{m}=I_{n}.$
Therefore, m=n.
So, A must be a square matrix.