Let `A` be a `2xx3` matrix, whereas `B` be a `3xx2` amtrix. If `det.(AB)=4`, then the value of `det.(BA)` is

Options

A. `-4`
B. `2`
C. `-2`
D. `0`

Explanation

To solve the problem, we need to find the value of \( \det(BA) \) given that \( \det(AB) = 4 \).

### Step-by-Step Solution:

1. **Understand the Dimensions**:
- Matrix \( A \) is a \( 2 \times 3 \) matrix.
- Matrix \( B \) is a \( 3 \times 2 \) matrix.
- The product \( AB \) results in a \( 2 \times 2 \) matrix.
- The product \( BA \) results in a \( 3 \times 3 \) matrix.

2. **Use the Determinant Property**:
- There is a property of determinants that states:
\[ \det(AB) = \det(A) \cdot \det(B) \]
- However, since \( A \) is a \( 2 \times 3 \) matrix and \( B \) is a \( 3 \times 2 \) matrix, we cannot directly compute \( \det(A) \) or \( \det(B) \) as they are not square matrices.

3. **Relate \( \det(AB) \) and \( \det(BA) \)**:
- Another important property is:
\[ \det(BA) = \det(A) \cdot \det(B) \]
- But since \( A \) and \( B \) are not square matrices, we can use the fact that:
\[ \det(BA) = \det(AB) \quad \text{(if both products are square matrices)} \]

4. **Apply the Given Information**:
- We know from the problem statement that:
\[ \det(AB) = 4 \]
- Therefore, since \( \det(BA) \) is also equal to \( \det(AB) \):
\[ \det(BA) = 4 \]

5. **Conclusion**:
- Thus, the value of \( \det(BA) \) is:
\[ \boxed{4} \]