I am new to Linear Algebra and this question is giving me problems because I end up with more unknowns than equations when trying to determine the unknown matrix. This is what I am given:
Matrix $A$:
\begin{bmatrix} 1 & 3 & 2 \\ 0 & -1 & 1 \end{bmatrix}
Matrix $AB$:
\begin{bmatrix} 3 & 4 \\ -1 & 0 \end{bmatrix}
Matrix $A$ is $2\times3$ and matrix $AB$ is $2\times2$. So I know that Matrix $B$ needs to be $3\times2$. So I let $B$ be an arbitrary $3\times2$ matrix
\begin{bmatrix} a & b \\ c & d \\ e & f \end{bmatrix}
and then multiply it out. I end up with these four equations:
$a + 3c + 2e = 3;\quad b + 3d + 2f = 4;\quad -c + e = 0;\quad -d + f = 0$
After this I don't know how to continue as I have more unknowns than equations.
I tried googling and I found a different matrix question on this site. This Matrix problem doesn't have the same more unknowns than equations problem as this one.
I also attempted finding the inverse of $A$ and that was instantly shut down as a rectangular matrix does not have an inverse. I am wondering if it is possible to find the inverse of $AB$ and somehow work it out like that, but I don't know how that would work.
Any help will be appreciated.
Best Answer
You seem to think that there should be a single answer. Why? There are infinitely many solutions. In fact, for each $e$ and each $f$,$$B=\begin{pmatrix}-5e&4-5f\\1+e&f\\e&f\end{pmatrix}$$is a solution. How did I get it? Simply solving the system of linear equations that you mentioned.