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)

Hint:

Tip 1: Always remember the reversal law: the transpose of a product is the product of the transposes in reverse order.
Tip 2: Transpose is distributive over addition and subtraction but not over multiplication.

Answer 1

The Correct Option is D

Concept:

• Properties of Transpose of a Matrix:
• \( (A^T)^T = A \)
• \( (kA)^T = k A^T \)
• \( (A \pm B)^T = A^T \pm B^T \)
• \( (AB)^T = B^T A^T \) (Reversal Law)

Step 1: Evaluate property (i)
By the addition property of transposes, \( (A + B)' = A' + B' \).
This is True.

Step 2: Evaluate property (ii)
By property, \( (A - B)' = A' - B' \).
The given expression \( B' - A' \) is only true if \( A' = B' \).
This is False in general.

Step 3: Evaluate property (iii)
By the reversal law of transposes, \( (AB)' = B'A' \).
The given expression \( A'B' \) is only true if the matrices commute under transpose.
This is False in general.

Step 4: Evaluate property (iv)
Using the constant property and the reversal law:
\[ (k AB)' = k (AB)' = k (B'A') \] This matches the given property.
This is True.