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Suppose $E$ is an elementary $n \times n$-matrix. Prove that if $A$ is any $n\times n$-matrix and $E$ is any elementary matrix, then $EA$ is a matrix obtained by carrying out a single elementary row operation on $A$, and that $AE$ is a matrix obtained by carrying out a single elementary column operation on $A$.

So I can write out the matrices $A$ and $E$, and I can conceptually see why the problem statement is true, but I just don't know how to prove it. Can anyone please help me here?

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It is useful to use the following properties of matrix multiplication:

Let $P$ and $Q$ be arbitrary matrices of dimensions $m \times k$ and $k \times n$ respectively. Let $p_1^T\dots,p_m^T$ denote the rows of $P$. Similarly, let $q_1,\dots,q_n$ denote the columns of $Q$. Then we have $$ PQ = P\pmatrix{q_1,\dots,q_n} = \pmatrix{Pq_1,\dots,Pq_n}\\ PQ = \pmatrix{p_1^T\\ \vdots \\p_m^T}Q = \pmatrix{p_1^TQ\\ \vdots \\p_m^TQ} = $$

Also,

let $e_1,\dots,e_n$ denote the standard basis (column) vectors of $\Bbb R^n$. If $M$ is a $n \times k$ matrix, then Then $e_j^T M$ is the $j$th row of $M$. If $M$ is a $k \times n$ matrix, then $M e_j$ is the $j$th column of $M$.

Now, note that the elementary matrices can be written out in terms of the standard basis vectors. Good luck.

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