If $$A$$ is a $$3 \times 3$$ matrix and $$|A| = 2$$, then $$|3 \text{ adj}(|3A| \cdot A^2)|$$ is equal to

We need to find $$|3 \cdot \text{adj}(|3A| \cdot A^2)|$$ for a $$3 \times 3$$ matrix $$A$$ with $$|A| = 2$$.

First, $$|3A| = 3^3 |A| = 27 \times 2 = 54$$. So the matrix inside the adjoint is $$54 \cdot A^2$$, which is a scalar times a matrix.

Now, $$|54 A^2| = 54^3 \cdot |A|^2 = 54^3 \times 4$$. For a $$3 \times 3$$ matrix $$M$$, $$|\text{adj}(M)| = |M|^{n-1} = |M|^2$$. Therefore $$|\text{adj}(54A^2)| = |54A^2|^2 = (54^3 \times 4)^2 = 54^6 \times 16$$.

Finally, $$|3 \cdot \text{adj}(54A^2)| = 3^3 \cdot |\text{adj}(54A^2)| = 27 \times 54^6 \times 16$$.

Simplifying: $$54 = 2 \times 27 = 2 \times 3^3$$, so $$54^6 = 2^6 \times 3^{18}$$. Then $$27 \times 54^6 \times 16 = 3^3 \times 2^6 \times 3^{18} \times 2^4 = 3^{21} \times 2^{10}$$.

We can write this as $$3^{11} \times 3^{10} \times 2^{10} = 3^{11} \times 6^{10}$$.

The correct answer is Option D: $$3^{11} \cdot 6^{10}$$.