1. For each of the matrices below, compute the characteristic polynomial, find the real eigenvalues and bases for the corresponding eigenspaces. $$ \left. \begin{array}{r} \left(\begin{array}{rr} 7 & 5 -10 & -8 \end{array} ight) W \left(\begin{array}{rrr) -1 & -21 4 & 5 \end{array} ight) W \left(\begin{array}{rrr} -1 & 0& O -4 & 2 & -1 4 & 0 & 3 \end{array} ight) W \left(\begin{array}{rrr} -1 & 0 & 1 -7 & 2 & 5 3 & 0 & 1 \end{array} ight) W \left. \begin{array}{rrr} -7 & -5 1 16 & 17 \end{array} ight) \left(\begin{array}{rrr} 1 & -21 1 & 2 \end{array} ight) W 6 & 4 & -3 2 & 0 & 3 \end{array} ight) } \end{array} $$ 2. If $A$ is an $n \times n$ matrix prove that $\lambda$ is an eigenvalue of $A$ if and only if $\lambda$ is an eigenvalue of $A^{T}$. 3. Suppose $A$ is an invertible $n \times n$ matrix and that $\lambda$ is an eigenvalue of $A$. Prove that $\lambda eq 0$ and that $1 / \lambda$ is an eigenvalue of $A^{-1}$.CS.SD. 119