Let A be an n x n matrix over the set of all real numbers. Let B be a matrix obtained from A by swapping two rows. Which of the following statements is/are TRUE? A. The determinant of B is the negative of the determinant of A B. If A is invertible, then B is also invertible C. If A is symmetric, then B is also symmetric D. If the trace of A is zero, then the trace of B is also zero

GATE 2024 · Engineering Mathematics · Determinant · medium

Answer: Statements A and B are TRUE. The answer is A, B.

1. Verify Statement A: det(B) = -det(A): Swapping any two rows of A multiplies the determinant by -1. Therefore det(B) = -det(A). Statement A is TRUE.
2. Verify Statement B: Invertibility is preserved: If A is invertible, then det(A) != 0. Since det(B) = -det(A), we have det(B) != 0. Therefore B is also invertible. Statement B is TRUE.
3. Disprove Statement C: Symmetry not preserved: Counterexample: A = [[1,2],[2,3]] is symmetric. Swap rows: B = [[2,3],[1,2]]. Then B^T = [[2,1],[3,2]] != B. So B is NOT symmetric. Statement C is FALSE.
4. Disprove Statement D: Trace not necessarily preserved: Counterexample: A = [[1,0],[-1,0]] has trace = 1 + 0 = 1 (not zero for trace-zero test, but consider trace(A)=0 case). Let A = [[1,0],[-1,-1]], trace = 0. Swap rows: B = [[-1,-1],[1,0]], trace = -1 + 0 = -1 != 0. So D can be FALSE.