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GATE 2024 CS (CS2) – Question 47

Engineering Mathematics · Linear Algebra · 2 marks · Multiple select

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. The determinant of $B$ is the negative of the determinant of $A$
  2. If $A$ is invertible, then $B$ is also invertible
  3. If $A$ is symmetric, then $B$ is also symmetric
  4. If the trace of $A$ is zero, then the trace of $B$ is also zero

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Show answer and explanation

Correct answer: (A) The determinant of $B$ is the negative of the determinant of $A$; (B) If $A$ is invertible, then $B$ is also invertible

Explanation

Swapping two rows negates the determinant, so invertibility is preserved. Symmetry and trace are not preserved in general.