###### Exercise 6

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

If \(A\) is similar to \(B\text{,}\) explain why \(B\) is similar to \(A\text{.}\)

If \(A\) is similar to \(B\) and \(B\) is similar to \(C\text{,}\) explain why \(A\) is similar to \(C\text{.}\)

If \(A\) is similar to \(B\) and \(B\) is diagonalizable, explain why \(A\) is diagonalizable.

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

If \(A\) and \(B\) are similar, explain why \(A\) and \(B\) have the same eigenvalues.