Let's analyze the problem carefully:

We are given that:

• Matrix $A$ has integer entries and its determinant is $\text{det}(A) = -4\sqrt{3}$.
• Matrix $A$ can be factored into the product of two matrices $B$ and $C$, where:
  • $B$ is a matrix with integer entries.
  • $C$ is a matrix whose entries are congruent to the identity matrix modulo 3 (i.e., $C \equiv I \mod 3$).
• We need to find $\det(B) \mod 3$.

Step 1: Determinant relationships

The determinant of a product of matrices satisfies the property: $\det(A) = \det(B) \cdot \det(C).$

Thus, we know: $\det(A) = \det(B) \cdot \det(C).$

Step 2: Considering $C \mod 3$

Since $C \equiv I \mod 3$, this means that: $\det(C) \equiv \det(I) = 1 \mod 3.$

Hence, $\det(C) \mod 3 = 1$.

Step 3: Solve for $\det(B) \mod 3$

Using the relation $\det(A) = \det(B) \cdot \det(C)$, and knowing that $\det(A) = -4\sqrt{3}$, we first need to understand the determinant modulo 3.

However, the determinant of $A$, which is $-4\sqrt{3}$, is not an integer. To handle this, the problem likely implies a reduction modulo 3 that only focuses on the integer part. Therefore, we reduce $\det(A) \mod 3$. Let's look at $-4 \mod 3$: $-4 \equiv 2 \mod 3.$

Thus, we conclude: $\det(A) \equiv 2 \mod 3.$

Finally, from the equation $\det(A) = \det(B) \cdot \det(C) \mod 3$, and knowing that $\det(C) \equiv 1 \mod 3$, we get: $\det(B) \equiv \det(A) \cdot \det(C)^{-1} \mod 3.$

Since $\det(C) = 1 \mod 3$, we find: $\det(B) \equiv 2 \mod 3.$

Thus, $\det(B) \mod 3 = 2$.

Would you like further clarification on any steps?

Here are 5 related questions to deepen understanding:

1. How does the determinant of a product of matrices relate to the determinants of the individual matrices?
2. What are the implications of reducing determinants modulo a number, and how does that affect matrix properties?
3. How do you calculate the inverse of a matrix modulo a prime number?
4. What other modulus operations could change the result of this determinant calculation?
5. How would the solution differ if the matrix $C$ had entries congruent to a different matrix modulo 3?

Tip: When solving problems involving determinants and modular arithmetic, always remember to apply matrix properties (like the product rule for determinants) and simplify step-by-step.