QuestionFind the characteristic equation, the eigenvalues, and bases for the eigenspaces of the following matrix: $A=\left[\begin{array}{ccc} 8 & 4 & -11 \\ 0 & -3 & 0 \\ 1 & 0 & -4 \end{array}\right]$
The characteristic equation is $\square$ $=0$ NOTE: Enter the eigenvalues in increasing order.
Eigenvalues $\lambda=$ $\square$ $\lambda=$ $\square$ $\square$ Choose one Choose one -

Studdy Solution
The characteristic equation is $\lambda^3 - \lambda^2 - 33\lambda - 63 = 0$ .
The eigenvalues are $\lambda_1 = -3$ , $\lambda_2 = -3$ , and $\lambda_3 = 7$ .
The eigenspace for $\lambda = -3$ is spanned by $\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$ and the eigenspace for $\lambda = 7$ is spanned by $\begin{bmatrix} 11 \\ 0 \\ 1 \end{bmatrix}$ .

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