Answer
Non-square matrices cannot have a multiplicative inverse since their products are not equal to the same multiplicative identity matrix.
Work Step by Step
Suppose there is an $n\times n$ square matrix A. If there exists another $n\times n$ matrix ${{A}^{-1}}$ such that:
$\begin{align}
& A{{A}^{-1}}={{I}_{n}} \\
& {{A}^{-1}}A={{I}_{n}}
\end{align}$
Then, the square matrix ${{A}^{-1}}$ is said to be a multiplicative inverse of the square matrix A. The result of multiplication of matrix A and ${{A}^{-1}}$ is an $n\times n$ square matrix, which is an identity matrix ${{I}_{n}}$.
Only square matrices can have multiplicative inverses. And also, it is not necessary that every square matrix will possess an inverse matrix.
Non-square matrices cannot have a multiplicative inverse.
This is because of the following reason:
A non-square matrix is a matrix that has a different number of rows and columns.
If A is a $m\times n$ matrix and B is a $n\times m$ matrix ( $n\ne m$ ), then the products AB and BA will be of different orders. This means two products AB and BA are not equal to the same multiplicative identity matrix.
