If A is a 3 × 3 matrix and A = 2 , then 3 adj 3 A A 2 is equal to

We need to find the value of $\left|3adj\left(\left|3A\right|A^2\right)\right|$

We know that $\left|kA\right|=k^n\left|A\right|$ where $n$ is the order of the matrix and $k$ is a constant.

$\Rightarrow \left|3A\right|=3^3\left(2\right)$

$\Rightarrow \left|3adj\left(\left|3A\right|A^2\right)\right|=3^3\left|adj\left(\left(3^3·2\right)A^2\right)\right|$

We know that $adj\left(kA\right)=k^{n-1}adj\left(A\right)$

$=3^3\left|{\left(3^3·2\right)}^{2}adj\left(A^2\right)\right|$

Now we know that $\left|adjA\right|={\left|A\right|}^{n-1}$

$=3^3{\left({\left(3^3·2\right)}^2\right)}^3\left|adj\left(A^2\right)\right|$

$=3^3{\left(3^3·2\right)}^6{\left|A^2\right|}^{3-1}$

$=3^3{\left(3^3·2\right)}^6{\left(2\right)}^4$

$=2^{10}·3^{21}$

Hence, the required answer is $2^{10}·3^{21}$$=6^{10}·3^{11}$