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