Question

If A is a matrix of order 3x3 and $|adj(adj(adj A))|=12^{4}$, then the value of $|A^{-1} adj A| =$

• A. 1
• B. 12
• C. $2\sqrt{3}$
• D. $\sqrt{6}$

Hint:

For a matrix of order $n$, $|adj(adj A)| = |A|^{(n-1)^{2}}$. Count the nested adjoins to determine the exponent.

Answer 1

The Correct Option is C

Step 1: Concept
Determinant properties: $|adj A| = |A|^{n-1}$ and $|kA| = k^{n}|A|$. For $n=3$, $|adj(adj(adj A))| = |A|^{(3-1)^{3}} = |A|^{8}$.

Step 2: Meaning
We are given $|A|^{8} = 12^{4}$, which implies $|A| = (12^{4})^{1/8} = 12^{1/2} = \sqrt{12} = 2\sqrt{3}$.

Step 3: Analysis
We need to find $|A^{-1} adj A|$. Using $|XY| = |X||Y|$, this becomes $|A^{-1}| \cdot |adj A|$. Since $|A^{-1}| = 1/|A|$ and $|adj A| = |A|^{2}$ for $n=3$.

Step 4: Conclusion
$|A^{-1} adj A| = (1/|A|) \cdot |A|^{2} = |A|$. Therefore, the value is $2\sqrt{3}$. Final Answer: (C)