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For the given matrix \( A = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} \), the following are the results:

### 1. Eigenvalues and Eigenvectors of \( A \):
- Eigenvalues: \( \lambda_1 = 1 \), \( \lambda_2 = 3 \)
- Eigenvectors:
  - For \( \lambda_1 = 1 \), the eigenvector is \( \begin{pmatrix} 1 \\ 1 \end{pmatrix} \)
  - For \( \lambda_2 = 3 \), the eigenvector is \( \begin{pmatrix} -1 \\ 1 \end{pmatrix} \)

### 2. Eigenvalues and Eigenvectors of \( A^2 \) (where \( A^2 = A \times A \)):
- Eigenvalues: \( \lambda_1 = 1 \), \( \lambda_2 = 9 \)
- Eigenvectors:
  - For \( \lambda_1 = 1 \), the eigenvector is \( \begin{pmatrix} 1 \\ 1 \end{pmatrix} \)
  - For \( \lambda_2 = 9 \), the eigenvector is \( \begin{pmatrix} -1 \\ 1 \end{pmatrix} \)

### 3. Verification of Trace and Determinant:
- Trace of \( A \): \( \text{Tr}(A) = 4 \), which matches the sum of the eigenvalues \( 1 + 3 = 4 \).
- Determinant of \( A \): \( \det(A) = 3 \), which matches the product of the eigenvalues \( 1 \times 3 = 3 \).

This confirms that the trace is equal to the sum of the eigenvalues and the determinant is equal to the product of the eigenvalues.

Would you like further details or have any questions?

Here are five relative questions to explore further:
1. How do we compute eigenvalues for larger matrices?
2. What is the geometric significance of eigenvectors?
3. How do the eigenvalues of a matrix change if we perform matrix transformations?
4. Why does squaring a matrix affect its eigenvalues in the way observed here?
5. What are the properties of a matrix whose eigenvalues are all positive?

**Tip**: The determinant provides important information about the invertibility of a matrix. If the determinant is zero, the matrix is singular and non-invertible.

Let A = ( 2  -1 -1  2 ). Find the eigenvalues and eigenvectors of both A and A^2. Then verify that the trace is the sum of the eigenvalues and the determinant is the product of the eigenvalues.

This problem explores how to find the eigenvalues and eigenvectors of a matrix A and its square A^2. It also verifies that the trace of A is equal to the sum of the eigenvalues and the determinant is the product of the eigenvalues. Key concepts include linear algebra, matrices, eigenvalues, eigenvectors, determinants, and matrix transformations.

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