Finding an Eigenvector Worksheet with Answer Key

What's on the worksheet

1. The matrix A = \begin{bmatrix} 8 & 1 \\ -6 & 3 \end{bmatrix} has eigenvalues 5 and 6.
2. The matrix A = \begin{bmatrix} 10 & -6 & -6 \\ -6 & 10 & 6 \\ 9 & -9 & -5 \end{bmatrix} has eigenvalues 4, with algebraic multiplicity 2, and 7.
3. The matrix A = \begin{bmatrix} -1 & -3 & 3 \\ -4 & -1 & 4 \\ -4 & -3 & 6 \end{bmatrix} has eigenvalues 2, −1 and 3.
4. Let A = \begin{bmatrix} 1 & 2 & -1 \\ 2 & 1 & -1 \\ 6 & -6 & 2 \end{bmatrix}.
5. Use the geometry first, then the matrix.
6. Both matrices below are upper triangular with characteristic polynomial (3 - )²(1 - ):
7. Let v be an eigenvector of the n × n matrix A for the eigenvalue .