Exercise 6

We say that $A$ is similar to $B$ if there is a matrix $P$ such that $A = PBP^{-1}$.

1. If $A$ is similar to $B$, explain why $B$ is similar to $A$.

2. If $A$ is similar to $B$ and $B$ is similar to $C$, explain why $A$ is similar to $C$.

3. If $A$ is similar to $B$ and $B$ is diagonalizable, explain why $A$ is diagonalizable.

4. If $A$ and $B$ are similar, explain why $A$ and $B$ have the same characteristic polynomial; that is, explain why $\det(A-\lambda I) = \det(B-\lambda I)$.

5. If $A$ and $B$ are similar, explain why $A$ and $B$ have the same eigenvalues.