Answer
False
Work Step by Step
Let $A=\begin{bmatrix}
1 & 0 &0\\0&0&1
\end{bmatrix} $
and $B=\begin{bmatrix}
1 &0 \\0&0\\0&1
\end{bmatrix} $
then $AB=I_2$ with A is not a square matrix.
It is not necessary that if $A$ is a matrix, there exists a matrix $B$ with $AB=I_n$ for $A$ to be invertible.
Hence, the statement is false.
