Let \(A=\begin{bmatrix} 2 & 3 & -3 \\ 1 & 0 & -1 \\ 1 & 1 & -2 \end{bmatrix}\) and let \(B=\begin{bmatrix} 0 & 1 & 0 \\ 3 & 0 & 1 \\ 2 & 0 & 0 \end{bmatrix}\).

Each of these matrices has the same characteristic polynomial, \((2-\lambda )(1+\lambda )^2\). Which of the following is true? (Justify your answer.)

\(A\) is diagonalizable but \(B\) is not diagonalizable. \(A\) is not diagonalizable but \(B\) is diagonalizable. Both \(A\) and \(B\) are diagonalizable. Neither \(A\) nor \(B\) is diagonalizable.

Source

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Problem 3.3.26.