CadyMath

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

\(-18\)

\(4\)

\(5\)

\(\frac{1}{10}\)

#6. \(A=\begin{bmatrix} 8 & -10 & 3\\7 & 8 & -3\\6 & 9 & -7\\\end{bmatrix}\)

\(-497\)

\(-6\)

\(9\)

\(\frac{9}{2}\)

#7. \(A=\begin{bmatrix} 3 & 0\\0 & 3\\\end{bmatrix}\)

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

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

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

\(\lambda_{1}=3\)

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

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

\(\lambda_{1}=0\)

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

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

#9. \(A=\begin{bmatrix} 1 & 0 & 2\\-4 & 2 & -7\\-4 & 0 & -5\\\end{bmatrix}\)

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

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

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

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

#10. \(A=\begin{bmatrix} -7 & 8 & 8\\-14 & 19 & 16\\10 & -14 & -11\\\end{bmatrix}\)

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

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

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

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

#11. \(A=\begin{bmatrix} 5 & -3\\4 & -2\\\end{bmatrix}\)

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

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

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

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

#12. \(A=\begin{bmatrix} 2 & 4\\0 & 3\\\end{bmatrix}\)

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

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

\(\bold{v_1}=\begin{bmatrix} -2\\3\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} 0\\\frac{3}{2}\\\end{bmatrix}\)

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

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

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

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

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

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

#14. \(A=\begin{bmatrix} 0 & -\frac{2}{3} & \frac{2}{3}\\1 & -\frac{5}{3} & \frac{2}{3}\\-2 & \frac{4}{3} & -\frac{7}{3}\\\end{bmatrix}\)

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

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

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

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