GATE CS Engineering Mathematics Previous Year Questions

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?

The determinant of \(B\) is the negative of the determinant of \(A\) If \(A\) is invertible, then \(B\) is also invertible If \(A\) is symmetric, then \(B\) is also symmetric If the trace of \(A\) is zero, then the trace of \(B\) is also zero

Explanation

Answer: (A), (B)

(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