Let $A$ be a matrix with dimension $n\times n$ , $B$ is a matrix with dimension $m\times m$ then if $f : R \to R$ where $R$ is the set of matrix with dimension $m\times n$ and $f(C) = ACB$ is isomorphism linear map then $A$ , $B$ are regular matrix
Edit: If $f$ is isomorphism then $f^{-1}$ is isomorphism which equal to $(ACB)^{-1}$ so $A^{-1}$ and $B^{-1}$ exist then $A$ and $B$ are regular , right ?
The map $f$ can be viewed as $$M_{m\times n}\overset{\cdot B}{\longrightarrow}M_{m\times n}\overset{A \cdot}{\longrightarrow}M_{m\times n},$$ or $$M_{m\times n}\overset{A \cdot}{\longrightarrow}M_{m\times n}\overset{ \cdot B}{\longrightarrow}M_{m\times n},$$ Then, if $f$ is a linear isomorphism, then the two maps $"\cdot B"$ and $"A\cdot"$ are injective, which implies that the rows of $B$ are linearly independent and columns of $A$ are linearly independent, thus $A$ and $B$ are nonsingular.