##### Exercise6

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

1. If $$A$$ is similar to $$B\text{,}$$ explain why $$B$$ is similar to $$A\text{.}$$

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

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 $$\det(A-\lambda I) = \det(B-\lambda I)\text{.}$$

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

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