Board 2026 Class 12 Maths Q3 Solution: Which transpose properties are true

The question

Which of the following properties is/are true for two matrices of suitable orders ?

(i) $(A + B)' = A' + B'$

(ii) $(A - B)' = B' - A'$

(iii) $(AB)' = A'B'$

(iv) $(kAB)' = kB'A'$ (k is a scalar)

(A) (i) only

(B) (i), (ii) and (iii)

(C) (i) and (ii)

(D) (i) and (iv)

Exam answer

(i) $(A + B)' = A' + B'$ is true.

(ii) is false, since $(A - B)' = A' - B'$ ⁠.

Why?

The transpose turns rows into columns: the $(i, j)$ entry of $M'$ is the $(j, i)$ entry of $M$ ⁠. Adding and subtracting also work entry by entry, so a transpose can be taken term by term:

\begin{aligned} (A + B)' &= A' + B' \\ (A - B)' &= A' - B' \end{aligned}

$B' - A'$ is the negative of $A' - B'$ ⁠, so (ii) fails unless $A = B$ ⁠.

(iii) is false, since $(AB)' = B'A'$ ⁠.

Why?

For a product the order reverses (the reversal law):

(AB)' = B'A'

$A'B'$ is a different product in general, because matrix multiplication is not commutative. If $A$ is $2 \times 3$ and $B$ is $3 \times 2$ ⁠, then $(AB)'$ is $2 \times 2$ but $A'B'$ is $3 \times 3$ ⁠.

Watch out: the order reverses only for products, never for sums or differences.

(iv) is true: $(kAB)' = k(AB)' = kB'A'$ ⁠.

Why?

A scalar comes straight out of a transpose: $(kM)' = kM'$ ⁠. Take $M = AB$ ⁠, then use the reversal law.

Only (i) and (iv) are true.

Answer: (D) (i) and (iv)

Where this comes from

Chapter 3, Matrices: Properties of the transpose.

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Q3: Which transpose properties are true