##### Activity4.3.3

1. Find a diagonalization of $$A\text{,}$$ if one exists, when

\begin{equation*} A = \left[\begin{array}{rr} 3 \amp -2 \\ 6 \amp -5 \\ \end{array}\right] \text{.} \end{equation*}
2. Can the diagonal matrix

\begin{equation*} A = \left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp -5 \\ \end{array}\right] \end{equation*}

be diagonalized? If so, explain how to find the matrices $$P$$ and $$D\text{.}$$

3. Find a diagonalization of $$A\text{,}$$ if one exists, when

\begin{equation*} A = \left[\begin{array}{rrr} -2 \amp 0 \amp 0 \\ 1 \amp -3\amp 0 \\ 2 \amp 0 \amp -3 \\ \end{array}\right] \text{.} \end{equation*}
4. Find a diagonalization of $$A\text{,}$$ if one exists, when

\begin{equation*} A = \left[\begin{array}{rrr} -2 \amp 0 \amp 0 \\ 1 \amp -3\amp 0 \\ 2 \amp 1 \amp -3 \\ \end{array}\right] \text{.} \end{equation*}
5. Suppose that $$A=PDP^{-1}$$ where

\begin{equation*} D = \left[\begin{array}{rr} 3 \amp 0 \\ 0 \amp -1 \\ \end{array}\right],\qquad P = \left[\begin{array}{cc} \vvec_2 \amp \vvec_1 \end{array}\right] = \left[\begin{array}{rr} 2 \amp 2 \\ 1 \amp -1 \\ \end{array}\right] \text{.} \end{equation*}
1. Explain why $$A$$ is invertible.

2. Find a diagonalization of $$A^{-1}\text{.}$$

3. Find a diagonalization of $$A^3\text{.}$$

in-context