Suppose $$A = \left[\begin{array}{cc} -5 &-4\cr 8 &7 \end{array}\right].$$ Find an invertible matrix $P$ and a diagonal matrix $D$ so that $A = P D P^{-1}$. Use your answer to find an expression for $A^{5}$ in terms of $P$, a power of $D$, and $P^{-1}$ in that order.

$A^{5} =$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$ $\left[\Rule{0pt}{2.4em}{0pt}\right.$$\left]\Rule{0pt}{2.4em}{0pt}\right.$.