How would I find Matrix $B$ in the following equation? Let $A$ be a $4\times 3$ matrix. Consider matrix $B$ which is a pre-multiplier of matrix $A$, that is, $BA$. Find matrix $B$ if it performs the following elementary row operation on $A$
Multiplies the second row by 4.
I let $C$ be the product after multiplying the second row by $4$ and put it in the equation $BA=C$.
I let $A=\begin{matrix}a&b&c\\d&e&f\\g&h&i\\j&k&l\\\end{matrix}$
and $C=\begin{matrix}a&b&c\\4d&4e&4f\\g&h&i\\j&k&l\\\end{matrix}$
My next step I thought would be to rearrange $BA=C$ into $B=CA^{-1}$. However this requires finding the inverse of a non-square matrix. How would I go about this? and how would I use this to find other operations such as:
Adds twice row $3$ to row $4$, or Interchanges rows $1$ and $3$.
Also my knowledge of matrices goes as far as Gaussian Elimination, Determinants and Cofactor/Adjoint Matrices. So I may not understand anything more advanced than this unless you can explain it well.
 A: You have good ideas, but in this case it's more effective to simply guess at what $B$ should be directly.  Write $B = \left[ \begin{matrix} \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \\ \end{matrix} \right]$ so that $BA = C$ becomes
$$
\left[ \begin{matrix} \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \\ \end{matrix} \right]
\left[\begin{matrix}a&b&c\\d&e&f\\g&h&i\\j&k&l\\\end{matrix}\right]
=
\left[\begin{matrix}a&b&c\\4d&4e&4f\\g&h&i\\j&k&l\\\end{matrix}\right]
$$
Now deduce the entries of $B$, keeping in mind that this has to be true for ALL $a,b,c,d,e,f,g,h,i,j,k,l$.  For example, the top row of $B$, when multiplied with the first column of $A$, gives a linear combination of $a,d,g,$ and $j$, which must then equal entry $a$ in matrix $C$.  So what does this say about the linear combination we had?  Hence, what do the entries in the top row of $B$ have to be?
The exact same technique works for adding twice row 3 to row 4 or interchanging rows 1 and 3.  Just write out matrix $A$ as $a,b,c,d,e,f,g,h,i,j,k,l$, and then write what matrix $C$ has to be.  Then, fill in the values of $B$ to make it work.
