This problem should be simple enough to solve by hand, but I don't have any good approach.
Let $K$ be an algebraically closed field. Consider matrices $A_1, A_2, B_1, B_2 \in M_2(K)$. Find a criterion equivalent to the simultaneous similarity of $(A_1, A_2)$ and $(B_1, B_2)$, i.e. there exists some $X \in \mathrm{GL}_2(K)$ such that $X A_1 X^{-1} = A_2$ and $X B_1 X^{-1} = B_2$.
In this case, each matrix above has an eigenvalue decomposition, i.e. is diagonalisable.