1 markMCQSection A

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)

Q: 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)

**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.

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.

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

Marking Scheme

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

Hint

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

Quick Oral Answer

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

Analysis & Explanation

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

Concept — standard transpose laws

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

Checking each statement

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

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).

Common Mistakes

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

Interesting Facts

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

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

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Frequently Asked Questions

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

This follows from how matrix multiplication is defined element-wise: the (i,j)th entry of $AB$ 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 $B'A'$ .

Does transpose reverse order for addition too?

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