Question

Let \(A\) be an \(n \times n\) matrix over the set of all real numbers \(\mathbb{R}\). 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

Hint:

Understand how row operations affect matrix properties such as determinant, symmetry, and invertibility.

Answer 1

The Correct Option is A

Option (A): When two rows of a matrix are swapped, the determinant of the new matrix is the negative of the determinant of the original matrix. Hence, (A) is correct.
Option (B): Swapping two rows of an invertible matrix does not affect its invertibility because the determinant remains non-zero. Hence, (B) is correct.
Option (C): Swapping rows of a symmetric matrix does not guarantee symmetry, as the symmetry condition \(A[i][j] = A[j][i]\) may be violated. Hence, (C) is incorrect.
Option (D): The trace of a matrix is the sum of its diagonal elements, which is independent of row swaps. Hence, if the trace of \(A\) is zero, the trace of \(B\) remains zero. This makes (D) incorrect. Final Answer: \[ \boxed{\text{(A), (B)}} \]