GATE CS 2024 Set 2 | Question 47

MSQ · 2 marks

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?

  1. Option A: The determinant of \(B\) is the negative of the determinant of \(A\) (correct)
  2. Option B: If \(A\) is invertible, then \(B\) is also invertible (correct)
  3. Option C: If \(A\) is symmetric, then \(B\) is also symmetric
  4. Option D: If the trace of \(A\) is zero, then the trace of \(B\) is also zero

Answer

(A), (B)

Explanation

  • (A) True: swapping two rows multiplies the determinant by \(-1\), so \(\det B = -\det A\).
  • (B) True: if \(A\) is invertible, \(\det B = -\det A \ne 0\), so \(B\) is invertible too.
  • (C) False: \(\begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}\) is symmetric, but swapping its rows gives \(\begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}\), which is not.
  • (D) False: \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) has trace 0, but swapping its rows gives the identity matrix, whose trace is 2. A row swap moves entries onto and off the diagonal, so the trace can change.

Answer: A, B