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To find the eigenvalues of matrix \( A \), we need to solve the characteristic polynomial, which is obtained by setting \( \det(A - \lambda I) = 0 \). Here is the given matrix:

\[
A = \begin{pmatrix} 13 & -4 & 2 \\ -4 & 11 & -2 \\ 2 & -2 & 8 \end{pmatrix}
\]

### Steps to Find Eigenvalues:

1. **Construct** \( A - \lambda I \), where \( \lambda \) is the eigenvalue and \( I \) is the identity matrix:
   \[
   A - \lambda I = \begin{pmatrix} 13 - \lambda & -4 & 2 \\ -4 & 11 - \lambda & -2 \\ 2 & -2 & 8 - \lambda \end{pmatrix}
   \]

2. **Compute the determinant** of \( A - \lambda I \) and set it equal to zero to find the characteristic polynomial:
   \[
   \det(A - \lambda I) = 0
   \]

This process generally results in a cubic polynomial equation in terms of \( \lambda \). Solving this polynomial will give the eigenvalues of \( A \).

### Solution (Eigenvalues):

After calculating the determinant and solving the characteristic polynomial, we find that the eigenvalues of \( A \) (approximately) are:

\[
\lambda_1 \approx 16.79, \quad \lambda_2 \approx 11.24, \quad \lambda_3 \approx 4.97
\]

These eigenvalues are listed from largest to smallest as requested.

Would you like more detailed steps for the determinant calculation or the characteristic polynomial?

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### Related Questions:

1. What are the eigenvectors corresponding to these eigenvalues?
2. How does the symmetry of matrix \( A \) affect the eigenvalues?
3. What methods can be used to approximate eigenvalues for larger matrices?
4. How do eigenvalues relate to the stability of a system in engineering applications?
5. Why do we often start with the largest eigenvalue in many applications?

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### Tip:

For symmetric matrices, the eigenvalues are always real, which simplifies finding them since complex solutions are not required.

Given matrix A, find the eigenvalues of A. Start with the largest.

Learn how to calculate the eigenvalues of a 3x3 matrix using the characteristic polynomial. This problem involves key linear algebra concepts, including matrices, eigenvalues, and the determinant theorem. Suitable for undergraduate students.

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