Let A be any skew-symmetric matrix (where \(A^T\) is Transpose of matrix A), then which of the following statements are correct?
(A) \(A^2\) is a symmetric matrix
(B) \(A^2\) is a skew-symmetric matrix
(C) \(A^TA = -A^2\)
(D) \(A^TA - AA^T = O\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
For a skew-symmetric matrix, \(A^T = -A\). Each statement follows from this and the rule \((PQ)^T = Q^TP^T\).
(A) \(A^2\) is a symmetric matrix — Correct. \((A^2)^T = A^TA^T = (-A)(-A) = A^2\).
(B) \(A^2\) is a skew-symmetric matrix — Incorrect. \((A^2)^T = A^2\), not \(-A^2\) (unless \(A^2 = O\)).
(C) \(A^TA = -A^2\) — Correct. \(A^TA = (-A)A = -A^2\).
(D) \(A^TA - AA^T = O\) — Correct. \((-A)A - A(-A) = -A^2 + A^2 = O\).