## Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible.

## Problem 216

Let

\[A=\begin{bmatrix}

1 & 3 & 3 \\

-3 &-5 &-3 \\

3 & 3 & 1

\end{bmatrix} \text{ and } B=\begin{bmatrix}

2 & 4 & 3 \\

-4 &-6 &-3 \\

3 & 3 & 1

\end{bmatrix}.\]
For this problem, you may use the fact that both matrices have the same characteristic polynomial:

\[p_A(\lambda)=p_B(\lambda)=-(\lambda-1)(\lambda+2)^2.\]

**(a)** Find all eigenvectors of $A$.

**(b)** Find all eigenvectors of $B$.

**(c)** Which matrix $A$ or $B$ is diagonalizable?

**(d)** Diagonalize the matrix stated in (c), i.e., find an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$ or $B=PDP^{-1}$.

(*Stanford University Linear Algebra Final Exam Problem*)

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