I think that I am close to having the right idea to finish off this problem, but I just want to clarify that the method that I am using is mathematically sound.
So, I have one $2\times2$ matrix (lets call it $D$) defined as \begin{bmatrix}a&b\\0&c\end{bmatrix} where $a$, $b$, and $c$ are numbers, and another $2\times2$ matrix (lets call it $N$) defined as \begin{bmatrix}x&y\\0&z\end{bmatrix} where $x$, $y$, and $z$ are unknown variables. The question is to solve for $x$, $y$, and $z$ such that $D$ times $N$ is equal to the identity matrix. What I did was multiply the two matrices together to get the matrix \begin{bmatrix}ax&ay+bx\\0&cz\end{bmatrix} and then set $ax=1$, $ay+bx=0$, and $cz=1$, and then solving for $x$, $y$, and $z$. I'm really just concerned with setting each element in the matrix product equal to their respective elements in the identity matrix. Is that fair game? Or is there something else I need to do to solve for $x$, $y$, and $z$?