Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We denote by $G L _ { r } ( A )$ the set of matrices of $M _ { r } ( A )$ that satisfy the equivalent properties of question III.3.d). a) Show that matrix multiplication induces a group structure on $G L _ { r } ( A )$. b) We define a relation on $M _ { s , r } ( A )$ by $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$. Show that this is an equivalence relation.
Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We denote by $G L _ { r } ( A )$ the set of matrices of $M _ { r } ( A )$ that satisfy the equivalent properties of question III.3.d).\\
a) Show that matrix multiplication induces a group structure on $G L _ { r } ( A )$.\\
b) We define a relation on $M _ { s , r } ( A )$ by $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$. Show that this is an equivalence relation.