If A and B are symmetric matrices, then which statements are correct ?
(A) (A − B)' = B' − A'
(B) (AB + BA) is symmetric matrix
(C) (AB)' = B'A'
(D) A'B' = B'A'
(E) (AB − BA) is skew symmetric matrix
Choose the correct answer from the options given below :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B), (C) and (E) Only
Since A and B are symmetric:
\(A' = A,\quad B' = B\)
The reversal law for transposes gives
\((AB)' = B'A' = BA\)
\((BA)' = A'B' = AB\)
So (AB)' = BA, (BA)' = AB and (A − B)' = A − B; AB and BA need not be equal.
Checking the statements
(A) (A − B)' = B' − A' — Incorrect. (A − B)' = A' − B' = A − B, while B' − A' = B − A.
(B) (AB + BA) is symmetric matrix — Correct. (AB + BA)' = BA + AB, the same matrix.
(C) (AB)' = B'A' — Correct. This is the reversal law, true for any two matrices that can be multiplied.
(D) A'B' = B'A' — Incorrect. It says AB = BA, which is not true for all symmetric matrices.
(E) (AB − BA) is skew symmetric matrix — Correct. (AB − BA)' = BA − AB = −(AB − BA).