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: {Use the determinant property of adjugate matrices}
For a square matrix \( A \) of order \( n \), the determinant of its adjugate is given by: \[ |{adj } A| = |A|^{n-1}. \] Since \( A \) is of order 3, we get: \[ |{adj } A^2| = |A^2|^{3-1} = |A^2|^2. \] Step 2: {Simplify further}
\[ |A^2| = (|A|^2), \] \[ |{adj } A^2| = (|A|^2)^2 = |A|^4. \] Step 3: {Compute \( |{Adj}({Adj } A^2)| \)}
\[ |{Adj}({Adj } A^2)| = (|A|^4)^{3-1} = (|A|^4)^2 = |A|^8. \] Step 4: {Conclusion}
Thus, the correct answer is \( |A|^8 \).