Exercise7

Suppose that \(A = PDP^{-1}\) where

\begin{equation*} D = \left[\begin{array}{rr} 1 \amp 0 \\ 0 \amp 0 \\ \end{array}\right],\qquad P = \left[\begin{array}{rr} 1 \amp -2 \\ 2 \amp 1 \\ \end{array}\right] \text{.} \end{equation*}
  1. Explain the geometric effect that \(D\) has on vectors in \(\real^2\text{.}\)

  2. Explain the geometric effect that \(A\) has on vectors in \(\real^2\text{.}\)

  3. What can you say about \(A^2\) and other powers of \(A\text{?}\)

  4. Is \(A\) invertible?

in-context