If $|A \text{ }adj B| \text{ }adj| (3A^{-1})|$, find the determinant.
Step-by-Step Solution
Key Concept: Use properties: $|adj(M)| = |M|^{n-1}$ and $|kM| = k^n|M|$ for $n \times n$ matrices
For $|A \text{ }adj B \text{ }adj|(3A^{-1})|$, we apply determinant properties: $|A| \times |adj B| \times |adj(3A^{-1})|$. Using $|adj(M)| = |M|^{n-1}$ for $n=3$: $|A| \times |B|^2 \times (3|A^{-1}|)^2 = |A| \times |B|^2 \times \frac{9}{|A|^2} = \frac{9|B|^2}{|A|}$. Substituting $|A|=x$, $|B|=x$ gives $\frac{9x^2}{x} = \frac{x^6}{27}$ when properly scaled.
Correct Answer: 1