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) \( (A\ B)^T=A^T\ B^T \)
(C) \( (A\,B\,C)^T=C^T\,B^T\,A^T \)
(D) \( (BA)^T=A^T\ B^T \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (C) and (D) only
The transpose of a sum is the sum of the transposes. For a product, the reversal law holds:
\( (AB)^T=B^TA^T \)
So \( (A+B)^T=A^T+B^T \), and the transpose of a product is the product of the transposes in reverse order.
Checking the statements
(A) \( (A+B)^T=A^T+B^T \) — Correct. The transpose of a sum is the sum of the transposes.
(B) \( (A\ B)^T=A^T\ B^T \) — Incorrect. The order must be reversed: \( (AB)^T=B^TA^T \).
(C) \( (A\,B\,C)^T=C^T\,B^T\,A^T \) — Correct. \( (ABC)^T=C^T(AB)^T=C^TB^TA^T \).
(D) \( (BA)^T=A^T\ B^T \) — Correct. The reversal law with B first.