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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

Explore the step-by-step solution to find eigenvalues and eigenvectors for the matrix A = [[2, 2], [5, -1]]. This guide uses the characteristic polynomial, determinant calculation, and quadratic formula, suitable for undergraduate students in linear algebra courses.

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