Row Switching Matrix I know that there exists an elementary matrix that switches the rows in another matrix when they are multiplied, but how do you prove that this elementary matrix actually does this job?

Let $I(i,j)$ be the identity matrix, except with the $i$th and $jth$ rows reversed. Then by definition of the matrix product, the $(k,l)$th component of the product $I(i,j)A$ is $$(I(i,j)A)_{k,l} = \sum_{m=1}^n (I(i,j))_{k,m}A_{m,l}.$$ If $k \neq i,j$, all the terms in the sum vanish except for the one corresponding to $m = k$, which is $(I(i,j))_{k,k}A_{k,l} = A_{k,l}$.
If $k = i$, say, then again all the terms in the sum vanish except for the one corresponding to $m = j$, which is $(I(i,j))_{k,j}A_{j,l} = A_{j,l}$. Similarly for the case $k = j$.
Thus we see that $(I(i,j)A)$ and $A$ coincide almost everywhere except that the rows $i$ and $j$ are swapped.
