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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?

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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.

Let A be a square matrix with integer entries such that its determinant is −4√3. If A can be expressed as a product of two matrices B and C, where B is a matrix with integer entries and C is a matrix with integer entries congruent to the identity matrix modulo 3, what is the value of det(B) modulo 3?

Explore a matrix factorization problem where the determinant of a matrix A is −4√3, and A can be expressed as a product of matrices B and C. Learn how to find det(B) modulo 3 using matrix and modular arithmetic techniques.

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