Exercises
B
Ex. 1
If $A$ is diagonalizable, then $A = P^{-1} D P$ for some diagonal matrix $D$. It follows \begin{aligned} A^T &= (P^{-1} D P)^T \\ &= P^T D^T (P^{-1})^T \\ &= Q D Q^{-1} \end{aligned} Where we defined $Q = P^T$.
Ex. 2
If $A$ is diagonal, then $A = P^{-1} D P$ for some diagonal matrix $D$. It follows \begin{aligned} A^{-1} &= (P^{-1} D P)^{-1} \\ &= P^{-1} D^{-1} P \\ &= P^{-1} D P \end{aligned}