Question

If \( A \) is a square matrix of order 3, then \( | {Adj}({Adj } A^2) | \) is:

• A. \( |A|^2 \)
• B. \( |A|^4 \)
• C. \( |A|^8 \)
• D. \( |A|^{16} \)

Hint:

For any square matrix \( A \) of order \( n \), we have: \[ |{Adj } A| = |A|^{n-1}. \]

Answer 1

The Correct Option is C

Step 1: {Apply the determinant property of adjugate matrices}
For an \( n \)-order square matrix \( A \), the determinant of its adjugate is calculated as \( |\text{adj } A| = |A|^{n-1} \). Given that \( A \) is of order 3, we have:\[|\text{adj } A^2| = |A^2|^{3-1} = |A^2|^2.\]Step 2: {Simplify the expression}
Since \( |A^2| = (|A|^2) \), the expression becomes:\[|\text{adj } A^2| = (|A|^2)^2 = |A|^4.\]Step 3: {Calculate \( |{\text{Adj}}({\text{Adj }} A^2)| \)}
Applying the adjugate determinant property again:\[|{\text{Adj}}({\text{Adj }} A^2)| = (|A|^4)^{3-1} = (|A|^4)^2 = |A|^8.\]Step 4: {Final Answer}
Therefore, the result is \( |A|^8 \).