If $A =\begin{bmatrix}0&0&\sqrt{3}\\0&\sqrt{3}&0\\\sqrt{3}&0&0\end{bmatrix}$ then $|adj\, A|$ is equal to

27

The correct answer is Option (3) → 27

$A=\begin{pmatrix} 0&0&\sqrt3\\ 0&\sqrt3&0\\ \sqrt3&0&0 \end{pmatrix}$

Compute determinant of $A$

$|A|=-\sqrt3\begin{vmatrix}0&\sqrt3\\ \sqrt3&0\end{vmatrix}$

$=-\sqrt3(0-3)$

$=3\sqrt3$

For a $3\times3$ matrix

$|\text{adj}\,A|=|A|^{2}$

$=(3\sqrt3)^2$

$=27$

The value of $|\text{adj}\,A|$ is $27$.