If A and B are symmetric matrices of same order, then which of the following are correct?
(A) $AB-BA$ is a skew-symmetric matrix.
(B) $AB+BA$ is a skew-symmetric matrix.
(C) $AB^T-BA^T$ is a skew-symmetric matrix.
(D) $AB+BA$ is a symmetric matrix.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (C) and (D) only **
Given: $A$ and $B$ are symmetric matrices.
So, $A^{T} = A$ and $B^{T} = B$.
(A) $AB - BA$ is skew-symmetric
Check transpose:
$(AB - BA)^{T} = B^{T}A^{T} - A^{T}B^{T} = BA - AB = -(AB - BA)$
Yes, skew-symmetric.
(B) $AB + BA$ is skew-symmetric
$(AB + BA)^{T} = BA + AB = AB + BA$
It equals itself → symmetric, not skew.
(C) $AB^{T} - BA^{T}$ is skew-symmetric
Since $A$ and $B$ are symmetric: $A^{T}=A$, $B^{T}=B$.
Expression becomes: $AB - BA$ (same as part A)
Skew-symmetric → TRUE.
(D) $AB + BA$ is symmetric
From part (B), it is symmetric → TRUE.
Correct statements: (A), (C), (D).