The photo you uploaded contains a matrix and some equations related to finding eigenvalues and eigenvectors. The matrix in the image is:

$A = \begin{bmatrix} 2 & 2 \\ 5 & -1 \end{bmatrix}$

Finding Eigenvalues

To find the eigenvalues, we compute the characteristic polynomial of the matrix $A$, given by the determinant of $A - \lambda I$, where $I$ is the identity matrix and $\lambda$ is a scalar (the eigenvalue). So, we solve:

$\text{det}(A - \lambda I) = 0$

$\text{det}\left(\begin{bmatrix} 2 - \lambda & 2 \\ 5 & -1 - \lambda \end{bmatrix}\right) = 0$

$(2 - \lambda)(-1 - \lambda) - (2 \cdot 5) = 0$

$\lambda^2 - \lambda - 2 - 10 = 0$

$\lambda^2 - \lambda - 12 = 0$

Using the quadratic formula:

$\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 1, b = -1, c = -12$, we can solve