Exercise 7

Suppose that $A = PDP^{-1}$ where $$D = \left[\begin{array}{rr} 1 & 0 \\ 0 & 0 \\ \end{array}\right],\qquad P = \left[\begin{array}{rr} 1 & -2 \\ 2 & 1 \\ \end{array}\right]$$.

1. Explain the geometric effect that $D$ has on vectors in $\real^2$.

2. Explain the geometric effect that $A$ has on vectors in $\real^2$.

3. What can you say about $A^2$ and other powers of $A$?

4. Is $A$ invertible?