If $A=\begin{bmatrix}1&2&1\\2&3&1\\0&0&1\end{bmatrix}$ then $|adj (3A^T)|^2$ is equal to

$3^{12}$

The correct answer is Option (3) → $3^{12}$

$A=\begin{pmatrix}1&2&1\\2&3&1\\0&0&1\end{pmatrix}$

Compute determinant of $A$

$|A|=1\begin{vmatrix}3&1\\0&1\end{vmatrix}-2\begin{vmatrix}2&1\\0&1\end{vmatrix}+1\begin{vmatrix}2&3\\0&0\end{vmatrix}$

$=1(3)-2(2)+0$

$=-1$

For a $3\times3$ matrix

$|\text{adj}(M)|=|M|^2$

Let $M=3A^T$

$|M|=|3A^T|=3^3|A^T|=27|A|=-27$

$|\text{adj}(3A^T)|=(-27)^2=3^6$

Hence

$|\text{adj}(3A^T)|^2=(3^6)^2=3^{12}$

The value of $|\text{adj}(3A^T)|^2$ is $3^{12}$.