If A=[\begin{matrix}0&3&4\\ -3&0&5\\ -4&-5&0\end{matrix}], then which one of the following statements is correct?
- A. A^{2} is symmetric matrix with det(A^{2})=0. ✓
- B. A^{2} is symmetric matrix with det(A^{2})\neq0.
- C. A^{2} is skew-symmetric matrix with det(A^{2})=0.
- D. A^{2} is skew-symmetric matrix with det(A^{2})\neq0.
Correct Answer: A. A^{2} is symmetric matrix with det(A^{2})=0.
Explanation
Matrix A is skew-symmetric, meaning A^T = -A. Taking the transpose of A^2 yields (A^2)^T = (A^T)^2 = (-A)^2 = A^2, so A^2 is symmetric. For any skew-symmetric matrix of order 3, det(A)=0, so det(A^2)=(det(A))^2=0.
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