CadyMath

#5. \(A=\begin{bmatrix} 10 & 4\\-7 & 8\\\end{bmatrix}\)

\(-23\)

\(108\)

\(18\)

\(23\)

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

\(-1\)

\(-3\)

\(0\)

\(3\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(\bold{v_1}=\begin{bmatrix} \frac{3}{2}\\0\\\end{bmatrix},\space\bold{v_1}=\begin{bmatrix} -1\\0\\\end{bmatrix}\)

#12. \(A=\begin{bmatrix} -\frac{11}{3} & -\frac{8}{3}\\\frac{2}{3} & -\frac{1}{3}\\\end{bmatrix}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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