For Problem U-4: Checking is easy: $A$v$_ 1 = \left[\begin{array}{rr}7&-4\ 2&1\end{array}\right]\left[\begin{array}{r}2\ 1\end{array}\right] = \left[\begin{array}{r}10\ 5\end{array}\right] = 5$   v$_ 1$ and $A$v$_ 2 = \left[\begin{array}{rr}7&-4\ 2&1\end{array}\right]\left[\begin{array}{r}1\ 1\end{array}\right] = \left[\begin{array}{r}3\ 3\end{array}\right] = 3$   v$_ 2$, so the corresponding eigenvalues are 5 and 3.

For Problem U-5: $A$v$= A($v$_ 1 +$   v$_ 2) = \left[\begin{array}{rr}7&-4\ 2&1\end{array}\right]\left[\begin{array}{r}3\ 1\end{array}\right] = \left[\begin{array}{r}17\ 7\end{array}\right]$, which is not a scalar times $\left[\begin{array}{r}3\ 1\end{array}\right]$.

For Problem U-6: This rotation can't have any eigenvectors because the result of rotating a nonzero vector v by $90 ^ \circ$ is never a scalar times v.