For Problem U-10: In the handout with pictures of a house, #1 to #1 is the identity matrix, which is scalar; #1 to #2 is the shear $\left[\begin{array}{rr}1&1\ 0&1\end{array}\right]$, which is defective; #1 to #3 is diagonal with distinct diagonal entries; #1 to #4 is a $90 ^ \circ$ rotation, so has no real eigenvalues; #1 to #5 is $\left[\begin{array}{rr}1&-1\ 1&1\end{array}\right]$, which has characteristic polynomial $\lambda ^ 2 + 2 \lambda + 2$ with roots $1 \pm i$ (from the quadratic formula), so no real eigenvalues--or notice that $\tau _ A$ rotates each vector by $45 ^ \circ$ and lengthens it, so no real vector is an eigenvector; #1 to #6 is diagonal with distinct diagonal entries; #1 to #7 is $\left[\begin{array}{rr}0&1\ 1&0\end{array}\right]$, symmetric but not scalar; #1 to #8 is $\left[\begin{array}{rr}3.5&-1.5\ 3&0\end{array}\right]$, which has characteristic polynomial $\lambda ^ 2 + 3.5 \lambda + 4.5$, which by the quadratic formula has no real roots, so there are no real eigenvalues.