Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We say that two matrices $M$ and $N$ of $M _ { s , r } ( A )$ are $A$-equivalent if $M \sim N$ (where $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$). If $M \in M _ { s , r } ( \mathbf { Z } )$ and $k$ is an integer at most equal to $\min ( r , s )$, we denote by $m _ { k } ( M )$ the gcd of the minors of size $k$ of $M$. Let $M$ and $N$ be two $\mathbf { Z }$-equivalent matrices of $M _ { s , r } ( \mathbf { Z } )$. Show that for all $k \leq \min ( r , s )$, we have $m _ { k } ( M ) = m _ { k } ( N )$ (one may begin by showing that $m _ { k } ( M )$ divides $m _ { k } ( N )$).
Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We say that two matrices $M$ and $N$ of $M _ { s , r } ( A )$ are $A$-equivalent if $M \sim N$ (where $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$). If $M \in M _ { s , r } ( \mathbf { Z } )$ and $k$ is an integer at most equal to $\min ( r , s )$, we denote by $m _ { k } ( M )$ the gcd of the minors of size $k$ of $M$.\\
Let $M$ and $N$ be two $\mathbf { Z }$-equivalent matrices of $M _ { s , r } ( \mathbf { Z } )$. Show that for all $k \leq \min ( r , s )$, we have $m _ { k } ( M ) = m _ { k } ( N )$ (one may begin by showing that $m _ { k } ( M )$ divides $m _ { k } ( N )$).