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The eigenvalues and eigenvectors of the matrix \( A = \begin{pmatrix} -59 & -18 \\ 180 & 55 \end{pmatrix} \) are as follows:

- Eigenvalues: \( -5 \) and \( 1 \)
- Eigenvectors corresponding to each eigenvalue (as columns of a matrix):
  \[
  \begin{pmatrix}
  -0.3162 & 0.2873 \\
   0.9487 & -0.9578
  \end{pmatrix}
  \]

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Here are 5 related questions:
1. How do we find the eigenvalues of a matrix in general?
2. What is the geometric significance of eigenvectors?
3. Can a matrix have complex eigenvalues and eigenvectors?
4. What role do eigenvalues play in determining the stability of systems?
5. How can eigenvalues help in matrix diagonalization?

**Tip**: Eigenvectors corresponding to distinct eigenvalues are always linearly independent.

Determine the eigenvalues and eigenvectors of the matrix A = [[-59, -18], [180, 55]].

Discover how to calculate the eigenvalues and eigenvectors of the matrix A = [[-59, -18], [180, 55]]. This problem covers key linear algebra concepts, including the characteristic equation and eigenvector determination. Suitable for undergraduate students.

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