Im stuck at this problem
Find an invertible Real Matrix $P$ such that $P^{-1}AP$ and $P^{-1}BP$ are both diagonal where $A$ and $B$ are real matrices.
a) $A=\begin{bmatrix} 1&2\\ 0&2\\ \end{bmatrix}$ and $B=\begin{bmatrix} 3&-8\\ 0&-1\\ \end{bmatrix}$
b) $A=\begin{bmatrix} 1&1\\ 1&1\\ \end{bmatrix}$ and $B=\begin{bmatrix} 1&a\\ a&1\\ \end{bmatrix}$
I know that diagonalisable commuting matrices can be simultaneously diagonalised but I'm not able to proceed.Kindly help.
Concerning the first pair of matrices, it turns out that the vectors $(1,0)$ and $(2,1)$ are eigenvectors of each of them. Therefore, take$$P=\begin{bmatrix}1&2\\0&1\end{bmatrix}.$$Can you solve the other problem now?