Question

Let \[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 4 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{bmatrix} \] Let det(A) and det(B) denote the determinants of the matrices A and B, respectively.
Which one of the options given below is TRUE?

• A. \( \text{det}(A) = \text{det}(B) \)
• B. \( \text{det}(B) = -\text{det}(A) \)
• C. \( \text{det}(A) = 0 \)
• D. \( \text{det}(AB) = \text{det}(A) + \text{det}(B) \)

Hint:

When two matrices are related by a row or column swap, their determinants are negatives of each other.

Answer 1

The Correct Option is B

The matrices \( A \) and \( B \) are related by a row swap operation. To find the determinant relationship between \( A \) and \( B \), observe the following:
- The matrix \( B \) is obtained by permuting the rows of matrix \( A \), which changes the sign of the determinant.
- Since \( B \) is a row permutation of \( A \), the determinant of \( B \) is the negative of the determinant of \( A \). Hence, \( \text{det}(B) = -\text{det}(A) \).
Thus, the correct answer is option (B).