The Characteristic Equation and Finding Eigenvalues Worksheet with Answer Key

What's on the worksheet

1. For each matrix, find the characteristic polynomial p() and every real eigenvalue with its algebraic multiplicity.
   \begin{bmatrix} 2 & 3 \\ 3 & -6 \end{bmatrix}\begin{bmatrix} 2 & -1 \\ 9 & 8 \end{bmatrix}\begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} 2 & -3 \\ 1 & 1 \end{bmatrix}
2. Let A = \begin{bmatrix} -1 & 2 \\ -3 & 4 \end{bmatrix}.
3. For a number k, let A = \begin{bmatrix} 2 & k & 0 \\ 1 & 3 & 0 \\ 4 & 5 & 1 \end{bmatrix}.
4. Find the characteristic polynomial of the matrix below in factored form, using a well-chosen expansion.
5. Every entry of the matrix below is nonzero.
6. Let U = \begin{bmatrix} 2 & 5 & -1 & 4 \\ 0 & -3 & 7 & 1 \\ 0 & 0 & 2 & 6 \\ 0 & 0 & 0 & 4 \end{bmatrix}. Give the eigenvalues, with their algebraic…
7. Let A be an n × n matrix and c ≠ 0 a number.