Let A be a square matrix of order `n xx n` where `n ge 2`. Let B be a matrix obtained from A with first and second rows interchanged. Then which one of the following is correct ?

Options

A. A. det A = det B
B. B. det A = - det B
C. C. A = B
D. D. A = - B

Answer

To solve the problem, we need to analyze the relationship between the determinants of matrix A and matrix B, where B is formed by interchanging the first and second rows of A.

### Step-by-Step Solution:

1. **Define the Matrix A**:
Let matrix A be a 2x2 matrix represented as:
\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

2. **Calculate the Determinant of A**:
The determinant of A (denoted as det(A)) is calculated as:
\[ \text{det}(A) = ad - bc \]

3. **Form the Matrix B**:
Matrix B is obtained by interchanging the first and second rows of matrix A:
\[ B = \begin{pmatrix} c & d \\ a & b \end{pmatrix} \]

4. **Calculate the Determinant of B**:
The determinant of B (denoted as det(B)) is calculated as:
\[ \text{det}(B) = cb - da \]

5. **Relate the Determinants**:
Notice that:
\[ \text{det}(B) = cb - da = -(ad - bc) = -\text{det}(A) \]
This shows that the determinant of B is the negative of the determinant of A.

6. **Conclusion**:
Therefore, we conclude that:
\[ \text{det}(A) = -\text{det}(B) \]
This corresponds to option B from the given choices.

### Final Answer:
The correct statement is:
\[ \text{det}(A) = -\text{det}(B) \]