To solve this problem, we need to analyze the properties of the $2 \times 2$ integer matrix $A$ that satisfies the conditions $A^2 = 3A$ and $\det(A) = 6$, then reduce it modulo 7 and determine the distinct values its trace can take.

Step 1: Using the equation $A^2 = 3A$

The equation $A^2 = 3A$ can be rewritten as: $A^2 - 3A = 0$ This implies that $A$ satisfies the polynomial $x^2 - 3x = 0$ or equivalently $x(x - 3) = 0$. Thus, the eigenvalues of $A$ must satisfy this polynomial equation, meaning the possible eigenvalues of $A$ are 0 and 3.

Step 2: Determining the trace of $A$

Since $A$ is a $2 \times 2$ matrix, the trace of $A$, denoted $\operatorname{tr}(A)$, is the sum of the eigenvalues. The possible eigenvalue pairs (since the eigenvalues are either 0 or 3) are:

1. $0$ and $0$ (trace = 0)
2. $3$ and $0$ (trace = 3)
3. $3$ and $3$ (trace = 6)

So, the possible values for the trace of $A$ are 0, 3, and 6.

Step 3: Condition on the Determinant

The determinant of $A$, denoted $\det(A)$, is the product of the eigenvalues. We are given that $\det(A) = 6$. This restricts the eigenvalues of $A$ because:

1. If the eigenvalues were $0$ and $0$, $\det(A) = 0$, which does not satisfy $\det(A) = 6$.
2. If the eigenvalues were $3$ and $0$, $\det(A) = 0 \times 3 = 0$, which also does not satisfy $\det(A) = 6$.
3. If the eigenvalues were $3$ and $3$, $\det(A) = 3 \times 3 = 9$, which also does not satisfy $\det(A) = 6$.

Thus, none of these eigenvalue combinations satisfies both $A^2 = 3A$ and $\det(A) = 6$.

Conclusion

There is no $2 \times 2$ integer matrix $A$ that satisfies both $A^2 = 3A$ and $\det(A) = 6$. Therefore, the problem has no solution, and there are no distinct values that the trace of the reduced matrix can take.

Let me know if you have any questions or want further clarification!

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Related Questions:

1. How do you determine the eigenvalues of a matrix given a polynomial equation?
2. What conditions must a matrix satisfy to have a specific determinant?
3. How does modulo arithmetic affect matrix properties such as trace and determinant?
4. What are the possible eigenvalues if a matrix satisfies a general equation $A^2 = kA$?
5. Can the trace and determinant conditions in this problem be satisfied if we relax one of the constraints?

Tip:

In problems involving matrices with polynomial constraints like $A^2 = kA$, analyzing eigenvalues often simplifies the conditions on the trace and determinant.