Exercise 3.1.12

Assuming $k=\min <!--\; FUNCTION\; APPLICATION\; -->\{m,n\}$. Set $j$ be a integer variable and do repeatly the following process:

• If $A_{\mathit{ij}}=0$ for all $j$, take $i=i+1$ and omit following steps and repeat process directly.
• If $A_{\mathit{ij}}\ne 0$ for some $j$, interchange the $i$-th and the $j$-th row.
• Adding $-\frac{A_{\mathit{ij}}}{A_{\mathit{ii}}}$ times the $i$-th row to the $j$-th row for all $j>i$.
• Set $i=i+1$ and repeat the process.