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GATE CSE 2024 Session 2 Q47: Properties preserved when two rows of a matrix are swapped

Question

Let AA be an n×nn \times n matrix over the set of all real numbers R\mathbb{R}. Let BB be a matrix obtained from AA by swapping two rows. Which of the following statements is/are TRUE?

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Answer: A, B (official GATE 2024 final answer key)

Concept used: A row swap is multiplication by a permutation matrix with determinant −1-1.

First step: Test (C) and (D) on the 2×22 \times 2 identity-like example A=(100−1)A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

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Source: official GATE 2024 Session 2 CS question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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