Question 61 markMCQSection A

If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric?

Matrices
Skew-Symmetric Matrices

Options

(A)ABAB
(B)AB+BAAB + BA
(C)(A+B)2(A + B)^2
(D)A−BA - B

Official answer

Correct option: (D) A - B


Since A′=−A and B′=−B, we get (A−B)′=A′−B′=−A−(−B)=−(A−B)A' = -A \text{ and } B' = -B, \text{ we get } (A - B)' = A' - B' = -A - (-B) = -(A - B). This means A−B satisfies the skew-symmetric condition M′=−M\text{This means } A - B \text{ satisfies the skew-symmetric condition } M' = -M.

Answer keywords:skew symmetric matrixA'=-Asymmetric matrixtranspose reversal lawmatrix combinationsAB and BA

Marking scheme

  1. 1.1 mark: correct option (D) selected — requires verifying the skew-symmetric condition using transpose reversal rules.

Hint

Use A′=−A, B′=−BA' = -A,\ B' = -B, and the reversal rule (XY)′=Y′X′(XY)' = Y'X' to test each option against the skew-symmetric condition M′=−MM' = -M.

Quick oral answer

Using A′=−A and B′=−BA' = -A \text{ and } B' = -B, (A−B)′ works out to −(A−B)(A-B)' \text{ works out to } -(A-B), satisfying the skew-symmetric condition, so A - B is the answer, option (D).

Common mistakes

  1. 1.Assuming AB is skew-symmetric just because both A and B individually are, without checking (AB)′=B′A′(AB)' = B'A' properly.
  2. 2.Mistaking AB+BAAB + BA for skew-symmetric when it is actually symmetric, since (AB+BA)′ simplifies back to AB+BA(AB+BA)' \text{ simplifies back to } AB+BA itself.
  3. 3.Forgetting to apply the reversal law (XY)′=Y′X′(XY)' = Y'X' when computing transposes of products, and instead writing (AB)′=A′B′(AB)' = A'B'.

Analysis and explanation

This tests the ability to apply the definitions A' = -A, B' = -B and the transpose reversal law (XY)' = Y'X' to combinations of skew-symmetric matrices.


Setup

  • Given: A′=−A and B′=−BA' = -A \text{ and } B' = -B (definition of skew-symmetric).

Checking (A - B)

  • (A−B)′=A′−B′=(−A)−(−B)=−A+B=−(A−B)(A - B)' = A' - B' = (-A) - (-B) = -A + B = -(A - B). This equals -(A-B), so A - B is skew-symmetric. Option (D) is correct.

Why the other options fail

  • (AB)': (AB)′=B′A′=(−B)(−A)=BA(AB)' = B'A' = (-B)(-A) = BA. For AB to be skew-symmetric we would need BA=−ABBA = -AB, which is not true in general (AB and BA are usually different matrices), so AB is not necessarily skew-symmetric.
  • (AB + BA)': (AB+BA)′=(AB)′+(BA)′=B′A′+A′B′=BA+AB=AB+BA(AB+BA)' = (AB)' + (BA)' = B'A' + A'B' = BA + AB = AB + BA. This equals the original matrix itself, so AB + BA is actually symmetric, not skew-symmetric.
  • ((A+B)²)': Let C=A+BC = A + B, so C′=A′+B′=−A−B=−CC' = A' + B' = -A - B = -C, meaning C is itself skew-symmetric. Then (C2)′=(C′)2=(−C)2=C2(C^2)' = (C')^2 = (-C)^2 = C^2, so C² is symmetric, not skew-symmetric.

Interesting facts

Every square matrix can be uniquely written as the sum of a symmetric matrix and a skew-symmetric matrix: A=12(A+A′)+12(A−A′)A = \frac{1}{2}(A+A') + \frac{1}{2}(A-A'), a decomposition used throughout physics (e.g., splitting stress tensors) and engineering.

All diagonal entries of a skew-symmetric matrix must be zero, since aii=−aii forces aii=0a_{ii} = -a_{ii} \text{ forces } a_{ii} = 0 — a quick way to instantly disqualify many matrices from being skew-symmetric.

Question details

Q6 of 44 questions

Board
CBSE
Class
Class 12
Subject
Mathematics
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Matrices
Topic
Skew-Symmetric Matrices
Bloom's level
analyze
Syllabus topic
Unit II: Algebra — Matrices: Symmetric and Skew-Symmetric Matrices
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

What is the definition of a skew-symmetric matrix?

A square matrix M is skew-symmetric if its transpose equals its negative, i.e., M′=−MM' = -M. This forces every diagonal entry to be zero.

Is the product of two skew-symmetric matrices always skew-symmetric?

No. If A and B are skew-symmetric, (AB)′=B′A′=(−B)(−A)=BA(AB)' = B'A' = (-B)(-A) = BA, which generally is not equal to -AB unless A and B commute in a special way. So the product AB is not skew-symmetric in general.