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Let's analyze the given information:

Let \( A \) be a \( 2 \times 2 \) matrix with integer entries such that:

1. \( A^2 = 3A \)
2. \( \det(A) = 6 \)

We need to find the distinct values the trace of the matrix \( A \) can take when reduced modulo 7.

### Step-by-Step Solution

1. **Matrix Equation \( A^2 = 3A \)**:
   This equation implies that \( A \) is similar to a matrix that satisfies a polynomial equation. Since \( A^2 = 3A \), we can rewrite this as:
   \[
   A^2 - 3A = 0 \quad \Rightarrow \quad A(A - 3I) = 0.
   \]
   This suggests that \( A \) could have eigenvalues that satisfy the equation \( \lambda^2 = 3\lambda \), leading to possible eigenvalues of \( \lambda = 0 \) or \( \lambda = 3 \).

2. **Determinant Condition \( \det(A) = 6 \)**:
   For a \( 2 \times 2 \) matrix with eigenvalues \( \lambda_1 \) and \( \lambda_2 \), the determinant is the product of the eigenvalues:
   \[
   \det(A) = \lambda_1 \cdot \lambda_2 = 6.
   \]
   Given the possible eigenvalues \( \lambda_1, \lambda_2 \in \{0, 3\}\), we need to check combinations that yield a product of 6, leading to integer eigenvalues like \( \lambda_1 = 2 \) and \( \lambda_2 = 3 \).

3. **Possible Values of the Trace**:
   The trace of \( A \) is the sum of its eigenvalues:
   \[
   \text{trace}(A) = \lambda_1 + \lambda_2.
   \]
   For eigenvalues \( \lambda_1 = 2 \) and \( \lambda_2 = 3 \), we find:
   \[
   \text{trace}(A) = 2 + 3 = 5.
   \]

4. **Reduce Modulo 7**:
   Now, we need to consider the trace modulo 7. Since the trace we found is 5, we observe:
   \[
   \text{trace}(A) \equiv 5 \pmod{7}.
   \]

### Conclusion
The trace of \( A \), when reduced modulo 7, can only take the distinct value **5**.

Would you like further details on any of these steps or have any questions?

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Here are five related questions for practice:

1. What are the possible eigenvalues of a matrix that satisfies \( A^2 = kA \) for a scalar \( k \)?
2. How does the trace of a matrix relate to its eigenvalues?
3. How do we compute the determinant of a \( 2 \times 2 \) matrix with known eigenvalues?
4. What are the properties of a matrix when the trace and determinant are known?
5. How does reducing a matrix modulo a prime affect its eigenvalues and trace?

**Tip**: In modular arithmetic problems involving matrices, reducing individual entries and results (like trace and determinant) modulo the required number can simplify the analysis significantly.

Let A be a 2x2 matrix with integer entries such that A^2 = 3A and det(A) = 6. Consider the matrix A reduced modulo 7. Determine how many distinct values the trace of the resulting matrix can take.

Solve for the distinct values the trace of a 2x2 matrix A can take modulo 7, given that A^2 = 3A and det(A) = 6. This problem involves linear algebra concepts like eigenvalues, determinants, and matrix reductions.

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