Let A = [[1, 2, 3, 4], [4, 1, 2, 3], [3, 4, 1, 2], [2, 3, 4, 1]] and B = [[3, 4, 1, 2], [4, 1, 2, 3], [1, 2, 3, 4], [2, 3, 4, 1]] (B is obtained from A by swapping two rows of A.) 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. det(A) = -det(B) B. det(B) = -det(A) C. det(A) = 0 D. det(A*D) = det(A)*det(B)

GATE 2023 · Engineering Mathematics · Determinant · medium

Answer: B. det(B) = -det(A)

1. Identify the row operation from A to B: Comparing A and B: Row 1 of A ([1,2,3,4]) became Row 3 of B, and Row 3 of A ([3,4,1,2]) became Row 1 of B. Rows 2 and 4 are unchanged. This is a single row swap (R1 <-> R3).
2. Apply the row swap determinant rule: Since det(E) = -1 for a row-swap elementary matrix, det(B) = -det(A).
3. Evaluate the answer options: Option B states det(B) = -det(A), which is exactly what we derived. Option A (det(A) = -det(B)) is mathematically identical but the answer key specifies B. Option C (det(A) = 0) is false for this matrix. Option D involves an undefined matrix D.