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246985038

Eigenvalues & EigenvectorsName __________

Compute the trace of matrix \(A\).

#1. \(A=\begin{bmatrix} 6 & -9\\5 & -6\\\end{bmatrix}\)

\(0\)

\(9\)

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

\(-3\)

\(-1\)

\(0\)

\(3\)

#3. \(A=\begin{bmatrix} 8 & -2\\-5 & 7\\\end{bmatrix}\)

\(-26\)

\(-4\)

\(-\frac{7}{9}\)

\(15\)

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

\(-11\)

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

\(0\)

\(11\)

#5. \(A=\begin{bmatrix} 9 & -7 & -75\\-4 & 8 & 48\\-10 & 5 & 75\\\end{bmatrix}\)

\(-92\)

\(-4\)

\(0\)

\(92\)

#6. \(A=\begin{bmatrix} -3 & 1 & 15\\-6 & 8 & 66\\-6 & -7 & -24\\\end{bmatrix}\)

\(-31\)

\(-19\)

\(1\)

\(19\)

Choose a set of correct eigenvectors of matrix \(A\).

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

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

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

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

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

#8. \(A=\begin{bmatrix} \frac{2}{5} & \frac{8}{5}\\\frac{12}{5} & -\frac{2}{5}\\\end{bmatrix}\)

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

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

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

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

#9. \(A=\begin{bmatrix} -\frac{23}{19} & \frac{60}{19}\\\frac{20}{19} & \frac{42}{19}\\\end{bmatrix}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Compute the eigenvalues of matrix \(A\).

#13. \(A=\begin{bmatrix} -\frac{7}{5} & -\frac{1}{5} & -\frac{1}{5}\\-\frac{2}{5} & -\frac{6}{5} & -\frac{1}{5}\\-\frac{4}{5} & -\frac{2}{5} & -\frac{7}{5}\\\end{bmatrix}\)

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

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

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

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

#14. \(A=\begin{bmatrix} 0 & -6 & -4\\-\frac{1}{5} & \frac{11}{5} & \frac{2}{5}\\-\frac{2}{5} & -\frac{18}{5} & -\frac{6}{5}\\\end{bmatrix}\)

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

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

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

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

Choose a set of correct eigenvectors of matrix \(A\).

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

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

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

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

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

#16. \(A=\begin{bmatrix} \frac{6}{5} & -\frac{16}{5}\\-\frac{4}{5} & -\frac{6}{5}\\\end{bmatrix}\)

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

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

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

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

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

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

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

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

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

#18. \(A=\begin{bmatrix} -\frac{3}{13} & \frac{12}{13}\\-\frac{4}{13} & \frac{16}{13}\\\end{bmatrix}\)

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

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

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

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

#19. \(A=\begin{bmatrix} 2 & 0\\0 & 2\\\end{bmatrix}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Compute the trace of matrix \(A\).

#23. \(A=\begin{bmatrix} -1 & 1\\-7 & -10\\\end{bmatrix}\)

\(-11\)

\(-\frac{5}{9}\)

\(0\)

\(17\)

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

\(-7\)

\(-6\)

\(-\frac{7}{8}\)

\(0\)

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