I just don't know how to start, so I'd like a push.

Let $\displaystyle A$ and $\displaystyle B$ be $\displaystyle 2\times 2$ matrices that satisfies $\displaystyle A^2B=3I$ and $\displaystyle A^TB^3=-I$, where $\displaystyle I$ is the identity matrix. Calculate $\displaystyle \det(A)$.