[A] is square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] – [A]T, respectively. Which of the following statements is TRUE? A. Both [S] and [D] are symmetric B. Both [S] and [D] are skew-symmetric C. [S] is skew-symmetric and [D] is symmetric D. [S] is symmetric and [D] is skew-symmetric

and [D] are symmetric” option2=”Both [S] and [D] are skew-symmetric” option3=”[S] is skew-symmetric and [D] is symmetric” option4=”[S] is symmetric and [D] is skew-symmetric” correct=”option1″]

The correct answer is: $\boxed{\text{C. }[S] \text{ is skew-symmetric and }[D] \text{ is symmetric}}$.

A square matrix $A$ is symmetric if $A^T = A$, and it is skew-symmetric if $A^T = -A$.

We are given that $A$ is neither symmetric nor skew-symmetric. This means that $A^T \neq A$ and $A^T \neq -A$.

The sum of two symmetric matrices is always symmetric. The sum of two skew-symmetric matrices is always skew-symmetric. The difference of a symmetric matrix and a skew-symmetric matrix is always skew-symmetric.

Therefore, $[S] = [A] + [A]^T$ is skew-symmetric, and $[D] = [A] – [A]^T$ is symmetric.