Question

If \(A\) is a square matrix of order \(n\) such that \(|\operatorname{adj}(\operatorname{adj} A)| = |A|^9\), then the value of \(n\) can be

• A. 4
• B. 2
• C. either 4 or 2
• D. None of these

Hint:

Use \(|\operatorname{adj} A| = |A|^{n-1}\) repeatedly.

Answer 1

The Correct Option is A

Step 1: Formula / Definition}
\[ |\operatorname{adj} A| = |A|^{n-1} \]
Step 2: Calculation / Simplification}
\(|\operatorname{adj}(\operatorname{adj} A)| = |\operatorname{adj} A|^{n-1} = (|A|^{n-1})^{n-1} = |A|^{(n-1)^2}\)
Given: \(|A|^{(n-1)^2} = |A|^9\)
\((n-1)^2 = 9 \Rightarrow n-1 = \pm 3\)
\(n-1 = 3 \Rightarrow n = 4\)
\(n-1 = -3 \Rightarrow n = -2\) (not possible)
\(\therefore n = 4\)
Step 3: Final Answer
\[ 4 \]