QuestionFind the eigenvalues and eigenvectors associated with the following matrix: $$A=\left[\begin{array}{cc} 3 & 5 \\ -1 & -3 \end{array}\right]$$

Studdy Solution
Find eigenvectors for each eigenvalue:
For $\lambda_1 = 2$:
Solve $(A - 2I)\mathbf{v} = \mathbf{0}$:
$A - 2I = \begin{bmatrix} 1 & 5 \\ -1 & -5 \end{bmatrix}$
Set up the system of equations:
$1v_1 + 5v_2 = 0$
$-v_1 - 5v_2 = 0$
Both equations are equivalent, so we can solve:
$v_1 = -5v_2$
Let $v_2 = t$, then $v_1 = -5t$.
Thus, an eigenvector is:
$\mathbf{v_1} = \begin{bmatrix} -5 \\ 1 \end{bmatrix}$
For $\lambda_2 = -2$:
Solve $(A + 2I)\mathbf{v} = \mathbf{0}$:
$A + 2I = \begin{bmatrix} 5 & 5 \\ -1 & -1 \end{bmatrix}$
Set up the system of equations:
$5v_1 + 5v_2 = 0$
$-v_1 - v_2 = 0$
Both equations are equivalent, so we can solve:
$v_1 = -v_2$
Let $v_2 = s$, then $v_1 = -s$.
Thus, an eigenvector is:
$\mathbf{v_2} = \begin{bmatrix} -1 \\ 1 \end{bmatrix}$
The eigenvalues are $\lambda_1 = 2$ and $\lambda_2 = -2$. The corresponding eigenvectors are $\mathbf{v_1} = \begin{bmatrix} -5 \\ 1 \end{bmatrix}$ and $\mathbf{v_2} = \begin{bmatrix} -1 \\ 1 \end{bmatrix}$.