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 \( \text{det}(BA) \) given that \( \text{det}(AB) = 4 \).

### Step-by-Step Solution:

1. **Identify the dimensions of matrices**:
- Matrix \( A \) is a \( 2 \times 3 \) matrix.
- Matrix \( B \) is a \( 3 \times 2 \) matrix.

2. **Determine the product \( AB \)**:
- The product \( AB \) will be a \( 2 \times 2 \) matrix because the number of columns in \( A \) (which is 3) matches the number of rows in \( B \) (which is also 3).
- Thus, \( AB \) is a square matrix, and we can compute its determinant.

3. **Use the property of determinants**:
- We know from the properties of determinants that:
\[ \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) \]
- However, since \( A \) is not a square matrix, we cannot directly apply this property.

4. **Relate \( \text{det}(AB) \) and \( \text{det}(BA) \)**:
- There is a useful property that relates the determinants of the products of matrices:
\[ \text{det}(AB) = \text{det}(BA) \]
- However, since \( AB \) is a \( 2 \times 2 \) matrix and \( BA \) is a \( 3 \times 3 \) matrix, we need to adjust for the dimensions.

5. **Apply the determinant property**:
- From the property of determinants, we also have:
\[ \text{det}(AB) = (-1)^{n} \text{det}(BA) \]
- Here, \( n \) is the number of rows of \( A \) (which is 2 in this case). Since \( n \) is even, we have:
\[ \text{det}(AB) = \text{det}(BA) \]
- Therefore, we can conclude:
\[ \text{det}(BA) = \text{det}(AB) \]

6. **Substituting the given value**:
- We know \( \text{det}(AB) = 4 \), so:
\[ \text{det}(BA) = 4 \]

### Final Answer:
\[ \text{det}(BA) = 4 \]