For matrices $A=[a_{ij}]_{m \times m}$ and $B=[b_{ij}]_{m \times m}$ where $A,B \in \mathbb{Z} ^{m \times m}$ let $A \equiv B \pmod{n}$ only if $a_{ij} \equiv b_{ij} \pmod{n}$ for every $i,j \in \{ 1,2,...,m \}$, that's $A-B=nZ$ for some $Z \in \mathbb{Z}^{m \times m}$. (The symbol $A \in \mathbb{Z} ^{m \times m}$ means that every element in $A$ is an integer.) Prove that for $A \in \mathbb{Z} ^{m \times m}$ there is $B \in \mathbb{Z} ^{m \times m}$ , where $AB \equiv I \pmod{n }$ only if $(\det (A),n)=1$ and find the value of $B$ in the form of $A$ where $I$ represents the dimensional identity matrix $m \times m$. (PP-nine)