Which of the following are true about the properties of transpose of appropriate matrices?
A. \( \mathrm{(A+B)^T=A^T+B^T} \)
B. \( \mathrm{(A-B)^T=A^T-B^T} \)
C. \( \mathrm{(A\,B)^T=A^T\,B^T} \)
D. \( \mathrm{(kA)^T=k(A^T)} \), where k is a scalar
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A, B and D only
The transpose of a sum or difference is the sum or difference of the transposes, and a scalar factor comes out unchanged. The transpose of a product reverses the order.
So \( (A\pm B)^T=A^T\pm B^T \) and \( (kA)^T=kA^T \), but \( (AB)^T=B^TA^T \).
Checking the statements
(A) \( \mathrm{(A+B)^T=A^T+B^T} \) — Correct. The transpose of a sum is the sum of the transposes.
(B) \( \mathrm{(A-B)^T=A^T-B^T} \) — Correct. The transpose of a difference is the difference of the transposes.
(C) \( \mathrm{(A\,B)^T=A^T\,B^T} \) — Incorrect. The order reverses: \( (AB)^T=B^TA^T \).
(D) \( \mathrm{(kA)^T=k(A^T)} \), where k is a scalar — Correct. A scalar factor comes out unchanged.