Question 31 markMCQSection A

Which of the following properties is/are true for two matrices of suitable orders? (i) (A+B)′=A′+B′(A + B)' = A' + B' (ii) (A−B)′=B′−A′(A - B)' = B' - A' (iii) (AB)′=A′B′(AB)' = A'B' (iv) (kAB)′=kB′A′(kAB)' = kB'A' (k is a scalar)

Matrices
Properties of Transpose of Matrices

Options

(A)(i) only
(B)(i), (ii) and (iii)
(C)(i) and (ii)
(D)(i) and (iv)

Official answer

Correct option: (D) (i) and (iv)


Only statements (i) and (iv) are true transpose identities. (ii) and (iii) misstate the standard reversal-order rules of transpose.

Answer keywords:transpose of matrix(AB)'=B'A'transpose propertiesreversal lawmatrix algebrascalar multiplication of matrix

Marking scheme

  1. 1.1 mark: correct option (D) selected — requires verifying all four statements correctly.

Hint

Remember: transpose of a product reverses order — (AB)′=B′A′(AB)' = B'A', not A′B′A'B'; transpose of addition/subtraction does NOT reverse order.

Quick oral answer

Transpose of a sum keeps the same order, A′+B′A'+B', but transpose of a product reverses order to B′A′B'A' — so only (i) and (iv) are correct, giving option (D).

Common mistakes

  1. 1.Assuming (AB)′=A′B′(AB)' = A'B' by analogy with addition, forgetting that matrix multiplication transpose reverses order.
  2. 2.Misremembering the subtraction rule and writing (A−B)′=B′−A′(A - B)' = B' - A' instead of the correct A′−B′A' - B'.
  3. 3.Not checking every one of the four statements individually and guessing based on partial recall.

Analysis and explanation

This checks precise recall and application of the standard transpose laws, especially the reversal rule for products.


Concept — standard transpose laws

  • (A+B)′=A′+B′(A + B)' = A' + B' — transpose distributes over addition.
  • (A−B)′=A′−B′(A - B)' = A' - B' — transpose also distributes over subtraction (note the order does NOT reverse for addition/subtraction).
  • (AB)′=B′A′(AB)' = B'A' — transpose of a product reverses the order of multiplication.
  • (kA)′=kA′(kA)' = kA' for scalar k — scalars pass through unchanged.

Checking each statement

  • (i) (A+B)′=A′+B′(A + B)' = A' + B' — TRUE, this is the standard addition law.
  • (ii) (A−B)′=B′−A′(A - B)' = B' - A' — FALSE, the correct identity is (A−B)′=A′−B′(A - B)' = A' - B', not B' - A' (sign is reversed incorrectly here).
  • (iii) (AB)′=A′B′(AB)' = A'B' — FALSE, the correct identity is (AB)′=B′A′(AB)' = B'A' (order must reverse, not stay the same).
  • (iv) (kAB)′=kB′A′(kAB)' = kB'A' — TRUE, since (AB)′=B′A′(AB)' = B'A' and scalars commute out unchanged, (kAB)′=k(AB)′=kB′A′(kAB)' = k(AB)' = kB'A'.

Why (D) is correct

  • Only (i) and (iv) hold true, matching option (D) exactly.

Why other options are wrong

  • (A) misses valid statement (iv).
  • (B) wrongly includes false statements (ii) and (iii).
  • (C) wrongly includes false statement (ii) and misses true statement (iv).

Interesting facts

The reversal rule (AB)′=B′A′(AB)' = B'A' extends to any number of matrices: (ABC)′=C′B′A′(ABC)' = C'B'A', a property heavily used in proving orthogonal matrix identities.

Transpose properties are foundational to defining symmetric (A′=AA' = A) and skew-symmetric (A′=−AA' = -A) matrices, which appear later in the very same paper.

Question details

Q3 of 44 questions

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

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Frequently asked questions

Why does (AB)′(AB)' equal B′A′B'A' and not A′B′A'B'?

This follows from how matrix multiplication is defined element-wise: the (i,j)th entry of ABAB is the dot product of row i of A and column j of B. Transposing swaps rows and columns, and to preserve the correct dimension compatibility and entry values, the order of multiplication must reverse, giving B′A′B'A'.

Does transpose reverse order for addition too?

No. Addition is commutative and transpose distributes over it directly without reversing order: (A+B)′=A′+B′=B′+A′(A + B)' = A' + B' = B' + A'. Only multiplication requires order reversal under transpose.