GATE CS 2024 Set 2 — Question 47
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Engineering Mathematics → Linear Algebra → Matrices & Determinants
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Question
Let be an matrix over the set of all real numbers . Let be a matrix obtained from by swapping two rows. Which of the following statements is/are TRUE?
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
Solution
Let's analyze each statement:(A) TRUE: A fundamental property of determinants is that swapping any two rows of a square matrix multiplies the determinant by . Therefore, .(B) TRUE: A matrix is invertible if and only if its determinant is non-zero. Since , if , then . Thus, if is invertible, is also invertible.(C) FALSE: Swapping rows generally destroys symmetry. For example, let , which is symmetric. Swapping the two rows results in . Since and , is not symmetric.(D) FALSE: The trace is the sum of the diagonal elements. Swapping rows changes which elements are on the diagonal. For example, let . Trace() . Swapping the rows gives . Trace() .
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