(Solved): In each of the following problems, determine whether the given matrix \( A \) is diagonalizable (j ...

In each of the following problems, determine whether the given matrix \( A \) is diagonalizable (justify your answer). If is \( A \) diagonalizable, find a matrix \( S \) such that \[ S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right) \] and identify the appropriate diagonal matrix given by \( \operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right) \). A. \( A=\left[\begin{array}{cc}1 & 2 \\ 2 & -2\end{array}\right] \) B. \( A=\left[\begin{array}{ccc}-1 & -2 & -2 \\ 2 & 3 & 2 \\ 0 & 0 & 1\end{array}\right] \) C. \( A=\left[\begin{array}{lll}0 & 3 & 3 \\ 3 & 0 & 3 \\ 3 & 3 & 0\end{array}\right] \)