Answer
-6 and -3 are the eigenvalues of A.
Work Step by Step
1. Calculate $Av_1$ and $Av_2$:
$$ Av_1 =
\begin{bmatrix} -5 & 2 \\ 1 & -4 \end{bmatrix}
\begin{bmatrix} -2 \\ 1 \end{bmatrix} =
\begin{bmatrix} 12 \\ -6 \end{bmatrix}
$$
$$ Av_2 =
\begin{bmatrix} -5 & 2 \\ 1 & -4 \end{bmatrix}
\begin{bmatrix} 1 \\ 1 \end{bmatrix} =
\begin{bmatrix} -3 \\ -3 \end{bmatrix}
$$
2. Find which number should multiply $v_1$ to result in $Av_1$ and the same for $v_2$ and $Av_2$
$$\begin{bmatrix} 12 \\ -6 \end{bmatrix} = -6 \begin{bmatrix} -2 \\ 1 \end{bmatrix} $$
$$
\begin{bmatrix} -3 \\ -3 \end{bmatrix} = -3\begin{bmatrix} 1 \\ 1 \end{bmatrix}
$$
Hint: Use the division operation between the numbers on the same position on the vectors.
3. Thus, -6 and -3 are the eigenvalues of A.
