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Beginning with an $n\times n$ matrix $A$, I would like to obtain a $m\times l$ submatrix, which need not be square. I'm aware of notation for submatrices, but are there any routine operations that can be used to algebraically obtain the submatrix? Might I be able to obtain matrices $B$ and $C$ so that $BAC$ gives the desired submatrix?

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Sure. Take the identity matrix and delete all rows except the ones you are interested in to get $B$. Then take the identity matrix and delete all columns except the ones you are interested in to get $C$.

For example, $$\begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}\begin{bmatrix}0 & 0 \\ 1 & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}e & f \\ h & i\end{bmatrix}$$

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