For Problem U-12: It was shown earlier that $A$ has eigenvalues $5,3$ and corresponding eigenvectors $\left[\begin{array}{r}2\ 1\end{array}\right]$ and $\left[\begin{array}{r}1\ 1\end{array}\right]$, so $P ^ {-1} A P = D$ for $P = \left[\begin{array}{rr}2&1\ 1&1\end{array}\right]$ and $D = \left[\begin{array}{rr}5&0\ 0&3\end{array}\right]$.