CadyMath

#5. \(A=\begin{bmatrix} -1 & -1\\-1 & -1\\\end{bmatrix}\)

\(-1\)

\(-2\)

\(-\frac{1}{7}\)

\(2\)

#6. \(A=\begin{bmatrix} 8 & -1 & -15\\2 & 6 & -10\\-2 & 10 & -6\\\end{bmatrix}\)

\(-7\)

\(-\frac{5}{8}\)

\(8\)

\(\frac{2}{7}\)

#7. \(A=\begin{bmatrix} 5 & -8\\4 & -7\\\end{bmatrix}\)

\(\lambda_{1}=-2,\space\lambda_{2}=0\)

\(\lambda_{1}=-3,\space\lambda_{2}=1\)

\(\lambda_{1}=1,\space\lambda_{2}=-1\)

\(\lambda_{1}=1,\space\lambda_{2}=2\)

#8. \(A=\begin{bmatrix} -\frac{16}{7} & \frac{4}{7}\\-\frac{8}{7} & \frac{2}{7}\\\end{bmatrix}\)

\(\lambda_{1}=-2,\space\lambda_{2}=-3\)

\(\lambda_{1}=0,\space\lambda_{2}=-2\)

\(\lambda_{1}=1,\space\lambda_{2}=-3\)

\(\lambda_{1}=2,\space\lambda_{2}=0\)

#9. \(A=\begin{bmatrix} \frac{5}{2} & 1 & 0\\\frac{1}{4} & \frac{5}{2} & 0\\-\frac{3}{2} & -3 & 0\\\end{bmatrix}\)

\(\lambda_{1}=-2,\space\lambda_{2}=1\)

\(\lambda_{1}=1,\space\lambda_{2}=-3\)

\(\lambda_{1}=1,\space\lambda_{2}=2\)

\(\lambda_{1}=2,\space\lambda_{2}=0,\space\lambda_{3}=3\)

#10. \(A=\begin{bmatrix} -23 & 42 & 32\\-28 & 51 & 40\\20 & -36 & -29\\\end{bmatrix}\)

\(\lambda_{1}=1,\space\lambda_{2}=-2,\space\lambda_{3}=-3\)

\(\lambda_{1}=2,\space\lambda_{2}=-2\)

\(\lambda_{1}=3,\space\lambda_{2}=-1,\space\lambda_{3}=-3\)

\(\lambda_{1}=3,\space\lambda_{2}=0,\space\lambda_{3}=1\)

#11. \(A=\begin{bmatrix} -\frac{2}{3} & \frac{4}{3}\\\frac{2}{3} & -\frac{4}{3}\\\end{bmatrix}\)

All vectors in \(\Reals^2\) are eigenvectors.

\(\bold{v_1}=\begin{bmatrix} -1\\0\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} -3\\-4\\\end{bmatrix}\)

\(\bold{v_1}=\begin{bmatrix} 3\\-1\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 1\\-1\\\end{bmatrix}\)

\(\bold{v_1}=\begin{bmatrix} \frac{3}{2}\\-\frac{3}{2}\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 2\\1\\\end{bmatrix}\)

#12. \(A=\begin{bmatrix} \frac{26}{11} & \frac{16}{11}\\-\frac{12}{11} & -\frac{26}{11}\\\end{bmatrix}\)

\(\bold{v_1}=\begin{bmatrix} -3\\4\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 2\\1\\\end{bmatrix}\)

\(\bold{v_1}=\begin{bmatrix} -4\\1\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 1\\-3\\\end{bmatrix}\)

\(\bold{v_1}=\begin{bmatrix} 1\\2\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} -1\\2\\\end{bmatrix}\)

\(\bold{v_1}=\begin{bmatrix} 1\\4\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 2\\-1\\\end{bmatrix}\)

#13. \(A=\begin{bmatrix} 0 & 0 & 0\\0 & 0 & 0\\1 & -1 & 1\\\end{bmatrix}\)

All vectors in \(\Reals^3\) are eigenvectors.

\(\bold{v_1}=\begin{bmatrix} -1\\0\\2\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 0\\1\\-1\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 0\\-1\\0\\\end{bmatrix}\)

\(\bold{v_1}=\begin{bmatrix} 0\\0\\1\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 1\\0\\-1\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 0\\1\\2\\\end{bmatrix}\)

\(\bold{v_1}=\begin{bmatrix} 0\\0\\1\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 1\\1\\0\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} -1\\1\\2\\\end{bmatrix}\)

#14. \(A=\begin{bmatrix} 2 & 0 & 0\\1 & 1 & 0\\-1 & -3 & -2\\\end{bmatrix}\)

All vectors in \(\Reals^3\) are eigenvectors.

\(\bold{v_1}=\begin{bmatrix} -1\\0\\-2\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} -1\\-1\\0\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} -1\\1\\0\\\end{bmatrix}\)

\(\bold{v_1}=\begin{bmatrix} 0\\-1\\1\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 1\\1\\-1\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 0\\0\\-1\\\end{bmatrix}\)

\(\bold{v_1}=\begin{bmatrix} 1\\1\\1\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} -1\\-2\\1\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 0\\-1\\-1\\\end{bmatrix}\)