Let A be a $3 \times 3$ matrix such that $|\text{adj}(\text{adj}(\text{adj}A))| = 12^4$. Then $|A^{-1}\text{adj}A|$ is equal to
Step-by-Step Solution
Key Concept: Use $|\text{adj}(\text{adj}(\text{adj}A))| = |A|^{(n-1)^3}$ for $n=3$, and $|A^{-1}\text{adj}A| = |A^{-1}||\text{adj}A| = \frac{1}{|A|} \cdot |A|^{n-1} = |A|^{n-2} = |A|$
Given $|\text{adj}(\text{adj}(\text{adj}A))| = 12^4$. For $n=3$: $|A|^{(n-1)^3} = |A|^8 = 12^4 \Rightarrow |A|^2 = 12 \Rightarrow |A| = 2\sqrt{3}$. Then $|A^{-1}\text{adj}A| = |A^{-1}| \cdot |\text{adj}A| = \frac{1}{|A|} \cdot |A|^{3-1} = |A| = 2\sqrt{3}$. Answer: (1)
Correct Answer: $2\sqrt{3}$