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?

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.