Multiple choice

[A] is a 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?

  1. Both [S] and [D] are symmetric.

  2. Both [S] and [D] are skew-symmetric.

  3. [S] is skew-symmetric and [D] is symmetric.

  4. [S] is symmetric and [D] is skew-symmetric.
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D Correct answer
Explanation

For any square matrix A, S = A + A^T is always symmetric because S^T = (A + A^T)^T = A^T + A = S. Conversely, D = A - A^T is always skew-symmetric because D^T = (A - A^T)^T = A^T - A = -(A - A^T) = -D.

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