Answer
True.
Work Step by Step
If A has a multiplicative inverse, $A^{-1},$ then
$AA^{-1}=I_{n}$ and $A^{-1}A=I_{n},$
where $I_{n}$ is an $n\times n$ identity matrix.
By definition of matrix multiplication,
from $AA^{-1}=I_{n},\ A$ must have n rows, and $A^{-1}$ must have n columns.
Also, from $ A^{-1}A=I_{n},\ A^{-1}$ must have n rows, and $A$ must have n columns.
So, in order to have a multiplicative inverse,
A must have n rows and n columns, that is,
be an n$\times$n matrix, a square matrix.
