Let A = [(3,5) (5,2)], then What is the trace of A^100?

The trace of a matrix is defined as the sum of the diagonal elements of the matrix. In this case, the matrix \( A = \begin{pmatrix} 3 & 5 \\ 5 & 2 \end{pmatrix} \).

### Step 1: Eigenvalues of \( A \)
The trace of a matrix is invariant under powers of the matrix, so the trace of \( A^n \) is the same as the trace of \( A \). This is due to the fact that the trace of a matrix equals the sum of its eigenvalues, and the eigenvalues of \( A^n \) are the eigenvalues of \( A \) raised to the power \( n \).

Let’s find the eigenvalues of matrix \( A \).

1. The characteristic equation is given by:
   \[
   \text{det}(A - \lambda I) = 0
   \]
   where \( I \) is the identity matrix and \( \lambda \) is the eigenvalue. Thus, we have:
   \[
   \text{det} \begin{pmatrix} 3-\lambda & 5 \\ 5 & 2-\lambda \end{pmatrix} = 0
   \]

2. Expanding the determinant:
   \[
   (3 - \lambda)(2 - \lambda) - 5 \cdot 5 = 0
   \]
   \[
   (6 - 5\lambda + \lambda^2) - 25 = 0
   \]
   \[
   \lambda^2 - 5\lambda - 19 = 0
   \]

3. Solving this quadratic equation using the quadratic formula:
   \[
   \lambda = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-19)}}{2(1)}
   \]
   \[
   \lambda = \frac{5 \pm \sqrt{25 + 76}}{2}
   \]
   \[
   \lambda = \frac{5 \pm \sqrt{101}}{2}
   \]

So, the eigenvalues of matrix \( A \) are:
\[
\lambda_1 = \frac{5 + \sqrt{101}}{2}, \quad \lambda_2 = \frac{5 - \sqrt{101}}{2}
\]

### Step 2: Trace of \( A^n \)
Since the trace of a matrix is the sum of its eigenvalues, and the trace remains constant for powers of a matrix:
\[
\text{Trace}(A^n) = \text{Trace}(A) = \lambda_1 + \lambda_2
\]
\[
\lambda_1 + \lambda_2 = \frac{5 + \sqrt{101}}{2} + \frac{5 - \sqrt{101}}{2} = \frac{5 + 5}{2} = 5
\]

Thus, the trace of \( A^n \) for any \( n \), including \( A^{100} \), is:
\[
\boxed{5}
\]

Would you like further clarification or details? Here are some related questions:

1. How are the eigenvalues of a matrix related to its powers?
2. Why does the trace of a matrix remain invariant under matrix powers?
3. Can the eigenvalues of a matrix be complex in some cases?
4. How would you compute the powers of a matrix directly?
5. How do determinant and trace relate in different matrices?

**Tip:** Eigenvalues are crucial in simplifying complex matrix operations, as they reveal essential characteristics like trace and determinant.

Discover how to find the trace of A^100 for the matrix A = [(3,5), (5,2)] by using eigenvalue techniques. This solution involves calculating eigenvalues, applying the characteristic equation, and understanding the trace invariance property of matrices. Ideal for undergraduate students studying linear algebra.

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