Definition. Let $A$ be an $r \times s$ integer valued matrix. $A$ is "special" if there exists an integer $k$ such that $a_{ij}=0$ unless $i=j$ and $i \leq k$ and $a_{ij} \neq 0$ if $i=j \leq k$.

Show that any matrix $A$ can be reduced to a "special" matrix by elementary row and column operations. A row operation is one of the following: 1. switching two rows. 2. multiplying one row by $-1$. 3. adding an integer multiple of a row to another row. Column operation is defined similarly.

I have no clue how to prove this claim. I can visualize intuitively that any matrix can be reduced to a matrix with non-zero entries in an upper part of the diagonal before row $k+1$ since we can first obtain the row reduced echelon form and subtract the extra column entries by column operations. But how do I prove that such procedure exists?


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