CadyMath

#5. \(A=\begin{bmatrix} 1 & -9\\-10 & -3\\\end{bmatrix}\)

\(-2\)

\(-\frac{1}{2}\)

\(2\)

\(\frac{1}{2}\)

#6. \(A=\begin{bmatrix} -3 & -1 & -9\\4 & 8 & 4\\-8 & 9 & -9\\\end{bmatrix}\)

\(-4\)

\(-580\)

\(3\)

\(8\)

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

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

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

\(\lambda_{1}=-3\)

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

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

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

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

\(\lambda_{1}=2\)

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

#9. \(A=\begin{bmatrix} -\frac{1}{5} & -\frac{4}{5} & -\frac{2}{5}\\-\frac{2}{5} & \frac{12}{5} & \frac{16}{5}\\\frac{2}{5} & -\frac{7}{5} & -\frac{11}{5}\\\end{bmatrix}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

#12. \(A=\begin{bmatrix} -\frac{24}{11} & -\frac{18}{11}\\-\frac{12}{11} & -\frac{9}{11}\\\end{bmatrix}\)

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

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

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

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

#13. \(A=\begin{bmatrix} -1 & -2 & 4\\0 & 3 & -6\\0 & 3 & -6\\\end{bmatrix}\)

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

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

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

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

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

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

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

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

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