GATE CS 2024 Set 2 Q.47: Linear Algebra

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

Official answer and solution

Official answer: A and B
From: Official GATE 2024 answer key

Solution

Use elementary row operation properties for determinants and rank, and test symmetry and trace with a $2 \times 2$ counterexample.

(A) is TRUE: Swapping two rows of a square matrix multiplies its determinant by $-1$ , so $\det(B) = -\det(A)$ .
(B) is TRUE: Since $\det(B) = -\det(A)$ , if $A$ is invertible ( $\det(A) \neq 0$ ), then $\det(B) \neq 0$ , so $B$ is also invertible.
(C) is FALSE: For the symmetric matrix $A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$ , swapping its two rows gives $B = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$ , which is not symmetric.
(D) is FALSE: For $A = \begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix}$ with $\text{tr}(A) = 1 - 1 = 0$ , swapping its two rows gives $B = \begin{pmatrix} 3 & -1 \\ 1 & 2 \end{pmatrix}$ with $\text{tr}(B) = 3 + 2 = 5 \neq 0$ .

Final answer

Options (A), (B)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 2 (CS2)
Question: 47 of 65
Section: Computer Science and Information Technology

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