If A and B are symmetric matrices of order 3 × 3 then the matrix $2AB - BA$ is:

nether symmetric nor skew-symmetric matrix

The correct answer is Option (4) → nether symmetric nor skew-symmetric matrix

Given $A$ and $B$ are symmetric matrices, so $A^{T}=A$ and $B^{T}=B$.

Let $M = 2AB - BA.$

Then, $M^{T} = (2AB - BA)^{T} = 2B^{T}A^{T} - A^{T}B^{T} = 2BA - AB.$

Hence, $M$ is neither symmetric nor skew-symmetric.

Correct answer: neither symmetric nor skew-symmetric matrix.