If $A=\begin{bmatrix}0&1&-3\\-1&0&5\\3&-5&0\end{bmatrix}$ then the value of $|A^{2025}|$ is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 0
Given
$A=\begin{pmatrix}0 & 1 & -3\\[4pt]-1 & 0 & 5\\[4pt]3 & -5 & 0\end{pmatrix}$
Check skew–symmetry:
$A'=\begin{pmatrix}0 & -1 & 3\\[4pt]1 & 0 & -5\\[4pt]-3 & 5 & 0\end{pmatrix}$
$-A=\begin{pmatrix}0 & -1 & 3\\[4pt]1 & 0 & -5\\[4pt]-3 & 5 & 0\end{pmatrix}$
Thus $A'=-A$, so $A$ is a skew–symmetric matrix.
A skew–symmetric matrix of odd order has determinant $0$.
$|A|=0$
$\Rightarrow |A^{2025}| = |A|^{2025} = 0^{2025} = 0$
$|A^{2025}| = 0$