How might I find matrix $M\in M_2(\mathbb C)$ such that $M^t A =M^{-1}$ where $A=\left[ \begin{array}{cc} a & a \\ a & a+1 \\ \end{array} \right]\in M_2(\mathbb N)$ without using the brute force method of writing $M=\left[ \begin{array}{cc} a & b \\ c & d \\ \end{array} \right]$? (I don't quite know how to solve the resulting system even in the brute force case anyway.)
Thanks in advance.