While I am studying, I found the statement that for $A \in M_{n}(K)$ : n-by-n square matrices with entries from some field K, if $rank(A)=r$, then there exist invertible matrices $U$ and $V$ in $M_{n}(K)$ such that

$A = U \begin{pmatrix}I_r &0\\ 0 &0\end{pmatrix} V$

(where $I_{r}$ is the r-by-r identity matrix).

Please help me how this works! It will be a great help! Thank you in advance!


Hint: The elementary row and column operation constitutes an invertible matrix for each operation. so apply them simultaneously to get $U$ and $V$ . I hope it helps.


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