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

Explanation

To solve the problem, we need to analyze the situation where we have a square matrix \( A \) of order \( n \times n \) and a matrix \( B \) obtained from \( A \) by interchanging the first and second rows. We want to determine the effect of this operation on the determinant of the matrix.

### Step-by-Step Solution:

1. **Understanding the Determinant**:
The determinant of a matrix is a scalar value that can be computed from its elements and provides important properties about the matrix, such as whether it is invertible.

2. **Effect of Row Interchange on Determinant**:
When two rows of a matrix are interchanged, the determinant of the new matrix is equal to the negative of the determinant of the original matrix. This is a fundamental property of determinants.

3. **Expressing the Relationship**:
Let \( \det(A) \) be the determinant of matrix \( A \). After interchanging the first and second rows to form matrix \( B \), we have:
\[ \det(B) = -\det(A) \]

4. **Conclusion**:
Therefore, if \( B \) is the matrix obtained from \( A \) by interchanging the first and second rows, then the correct statement is:
\[ \text{If } \det(A) \text{ is the determinant of } A, \text{ then } \det(B) = -\det(A). \]

### Final Answer:
The correct statement is that the determinant of matrix \( B \) is the negative of the determinant of matrix \( A \).

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