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.