Question

If A, B and C are square matrices of order n x n, then which of the following are TRUE? [Where $A^T$ is transpose of matrix A]
(A) \( (A + B)^T = A^T + B^T \)
(B) \( (AB)^T = A^T B^T \)
(C) \( (ABC)^T = C^T B^T A^T \)
(D) \( (BA)^T = A^T B^T \)
Choose the correct answer:

• A. (A), (B) and (C) only
• B. (A) and (D) only
• C. (A) and (B) only
• D. (A), (C) and (D) only

Hint:

Always remember the "reversal law" for matrix transposition: the order of matrices must be reversed when transposing a product.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The transpose operator \( (\cdot)^T \) follows specific properties:
1. The transpose of a sum is the sum of the transposes: \( (A+B)^T = A^T + B^T \).
2. The transpose of a product is the product of the transposes in REVERSE order: \( (AB)^T = B^T A^T \).

Step 2: Detailed Explanation:
- Property (A): \( (A + B)^T = A^T + B^T \) is TRUE.
- Property (B): \( (AB)^T = A^T B^T \) is FALSE because the correct property is \( (AB)^T = B^T A^T \).
- Property (C): \( (ABC)^T = ((AB)C)^T = C^T (AB)^T = C^T (B^T A^T) = C^T B^T A^T \) is TRUE.
- Property (D): \( (BA)^T = A^T B^T \) is FALSE as per property (B), however, looking at the provided choice structure for (D) in the question, note that the reversal law \( (BA)^T = A^T B^T \) is mathematically correct because the order of multiplication for \( (BA)^T \) is \( A^T B^T \).

Step 3: Final Answer:
Statements (A), (C), and (D) are true according to matrix transpose laws.