If A is a skew-symmetric matrix, then which of the following statements is NOT true?
(A) A is singular if order of A is odd
(B) A is non-singular
(C) $A^{2025}$ is a skew-symmetric matrix
(D) $A^{2025}$ is a symmetric matrix
(E) all diagonal elements of A are zeros
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (D) only
For a skew-symmetric matrix $A$, $A^T=-A$.
(A) True: If order of $A$ is odd, then $\det(A)=\det(A^T)=\det(-A)=(-1)^n\det(A)=-\det(A)$ ⟹ $\det(A)=0$. So $A$ is singular.
(B) False: A skew-symmetric matrix of odd order is singular, hence it cannot be non-singular.
(C) True: For any skew-symmetric matrix, odd powers remain skew-symmetric, since $(A^{2k+1})^T=(-A)^{2k+1}=-A^{2k+1}$.
(D) False: $A^{2025}$ is skew-symmetric, not symmetric.
(E) True: All diagonal elements of a skew-symmetric matrix are zero.
Statement (B) and Statement (D) is NOT true.