Question

If A is a square matrix of order 3, then |Adj(Adj A²)| =

• A. |A|²
• B. |A|⁴
• C. |A|⁸
• D. |A|¹⁶

Answer 1

The Correct Option is C

To solve the problem, we need to evaluate \( \left| \text{Adj}(\text{Adj}(A^2)) \right| \), where \( A \) is a square matrix of order 3.

1. Recall Key Properties:
Let \( A \) be an \( n \times n \) matrix. The following results are useful:

• \( \left| \text{Adj}(A) \right| = |A|^{n-1} \) if \( A \) is invertible.
• \( \text{Adj}(A^2) = \text{Adj}(A) \cdot \text{Adj}(A) \) only when \( A \) is invertible, and for determinant purposes, we use composition rules properly.

2. Step-by-Step Determinant Computation:
We are given \( A \) is a 3x3 matrix, i.e., \( n = 3 \).
Start with \( A^2 \). It’s also a 3x3 matrix, so: \[ \left| \text{Adj}(A^2) \right| = |A^2|^{n - 1} = |A|^{2(n - 1)} = |A|^{2 \cdot 2} = |A|^4 \] Now, apply the adjugate again: \[ \left| \text{Adj}(\text{Adj}(A^2)) \right| = \left| \text{Adj}(B) \right|, \text{ where } B = \text{Adj}(A^2), \text{ and } B \text{ is also a 3x3 matrix} \] \[ \Rightarrow \left| \text{Adj}(B) \right| = |B|^{n - 1} = (|A|^4)^{2} = |A|^8 \]

Final Answer:
The value of \( \left| \text{Adj}(\text{Adj}(A^2)) \right| \) is \( |A|^8 \)