Exercise 10

We say that two square matrices $A$ and $B$ are similar if there is an invertible matrix $P$ such that $B = PAP^{-1}$.

1. If $A$ and $B$ are similar, explain why $A^2$ and $B^2$ are similar as well. In particular, if $B = PAP^{-1}$, explain why $B^2 = PA^2P^{-1}$.

2. If $A$ and $B$ are similar and $A$ is invertible, explain why $B$ is also invertible.

3. If $A$ and $B$ are similar and both are invertible, explain why $A^{-1}$ and $B^{-1}$ are similar.

4. If $A$ is similar to $B$ and $B$ is similar to $C$, explain why $A$ is similar to $C$. To begin, you may wish to assume that $B = PAP^{-1}$ and $C = QBQ^{-1}$.