Which of the following statements are true?
(A) A matrix which is both symmetric as well as skew-symmetric is a null matrix.
(B) All positive integral powers of a symmetric matrix are symmetric.
(C) If A is a square matrix, then \( \mathrm{A}+\mathrm{A}^{\mathrm{T}} \) is a symmetric matrix.
(D) All positive even integral powers of a skew-symmetric matrix are skew-symmetric.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (C) only
All four statements follow from the rules of the transpose:
\( \displaystyle (A^n)^T=(A^T)^n \)
\( \displaystyle (A+B)^T=A^T+B^T,\ \ (A^T)^T=A \)
For a skew-symmetric A (\( A^T=-A \)) and an even power 2k:
\( \displaystyle (A^{2k})^T=(-A)^{2k}=A^{2k} \)
So a matrix with \( A^T=A \) and \( A^T=-A \) is O, powers of a symmetric matrix are symmetric, \( A+A^T \) is symmetric, and even powers of a skew-symmetric matrix are symmetric.
Checking the statements
(A) A matrix which is both symmetric as well as skew-symmetric is a null matrix. — Correct. \( A=-A \) gives \( 2A=O \), so \( A=O \).
(B) All positive integral powers of a symmetric matrix are symmetric. — Correct. \( (A^n)^T=(A^T)^n=A^n \).
(C) If A is a square matrix, then \( \mathrm{A}+\mathrm{A}^{\mathrm{T}} \) is a symmetric matrix. — Correct. \( (A+A^T)^T=A^T+A \).
(D) All positive even integral powers of a skew-symmetric matrix are skew-symmetric. — Incorrect. Even powers are symmetric; for example \( \begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}^2=-\mathrm{I} \), which is not skew-symmetric.