Let A be an n n matrix over the set of all real numbers R. Let B be a matrix obtained from A by swapping two rows. Which of the following statements is/are TRUE? A) The determinant of B is the negative of the determinant of A B) If A is invertible, then B is also invertible C) If A is symmetric, then B is also symmetric D) If the trace of A is zero, then the trace of B is also zero

Official 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: Let A be an n n matrix and B be obtained by swapping two rows of A. - Option A: By elementary row operations, interchanging two rows of a matrix negates the determinant. Thus, (B) = - (A). This statement is TRUE. - Option B: A matrix is invertible if and only if its determinant is non-zero. Since (B) = - (A), if (A) 0, then (B) 0. Therefore, if A is invertible, B is also invertible. This statement is TRUE. - Option C: Swapping two rows of a symmetric matrix does not necessarily yield a symmetric matrix. For example, consider the symmetric matrix: A = pmatrix 1 & 2 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 pmatrix Swapping row 1 and row 3 gives: B = pmatrix 0 & 0 & 1 \\ 2 & 1 & 0 \\ 1 & 2 & 0 pmatrix Here BT B, so B is not symmetric. This statement is FALSE. - Option D: Consider the matrix with trace 0: A = pmatrix 2 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 pmatrix Here trace(A) = 2 - 1 - 1 = 0. Swapping row 1 and row 3 gives: B = pmatrix 0 & 0 & -1 \\ 0 & -1 & 0 \\ 2 & 0 & 0 pmatrix The trace of B is 0 + (-1) + 0 = -1 0. This statement is FALSE. Hence, the correct statements are A and B.

A) The determinant of B is the negative of the determinant of A

B) If A is invertible, then B is also invertible

C) If A is symmetric, then B is also symmetric

D) If the trace of A is zero, then the trace of B is also zero

Engineering Mathematics · Linear Algebra

Official IIT answer key · IISc Bangalore · Audited Aug 2026

Take topic test

GATE CSE 2024 Set 2 Q47 · 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?

Select one or more options

Key concept

Linear Algebra

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Linear Algebra

Linear Algebra in GATE CS is consistently tested across fundamental properties of matrices, systems of linear equations, and spectral theory (eigenvalues and eigenvectors). Questions heavily leverage algebraic identities such as trace-sum, determinant-product relations, rank-nullity theorem, and structural properties like orthogonality of symmetric matrices and LU decomposition. Problems frequently appear as 1-mark and 2-mark conceptual MCQs/MSQs or direct numerical computation NATs.

Full Linear Algebra guide →

More on Linear Algebra

All 46 questions on Linear Algebra →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Engineering Mathematics · Linear Algebra

Official IIT answer key · IISc Bangalore · Audited Aug 2026

Take topic test

GATE CSE 2024 Set 2 Q47 · 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?

Select one or more options

Key concept

Linear Algebra

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Linear Algebra

Full Linear Algebra guide →

More on Linear Algebra

All 46 questions on Linear Algebra →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

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?

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2024 Set 2

More on 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?

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2024 Set 2

More on Linear Algebra

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

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2024 Set 2

More on Linear Algebra

Share this solution

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

Step-by-step verified solution

Check you can still solve Linear Algebra under a clock

More from GATE CSE 2024 Set 2

More on Linear Algebra

Share this solution

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?

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?