If `A` is an invertible matrix of order `3xx3` such that `|A|=2` . Then, find `a d j\ (a d j\ A)` .

Options

A. `abs(A)A`
B. `abs(A)^2A`
C. `abs(A)^(-1)A`
D. none of these

Explanation

To solve the problem, we need to find \( \text{adj}(\text{adj}(A)) \) given that \( A \) is an invertible matrix of order \( 3 \times 3 \) and \( |A| = 2 \).

### Step-by-Step Solution:

1. **Understanding the Properties of the Adjoint**:
The adjoint of a matrix \( A \), denoted as \( \text{adj}(A) \), has a specific relationship with the determinant of \( A \). For a square matrix \( A \) of order \( n \):
\[ A \cdot \text{adj}(A) = |A| I_n \]
where \( I_n \) is the identity matrix of order \( n \).

2. **Finding \( \text{adj}(A) \)**:
Since \( |A| = 2 \) and \( n = 3 \), we can express:
\[ A \cdot \text{adj}(A) = 2 I_3 \]

3. **Finding \( \text{adj}(\text{adj}(A)) \)**:
There is a property that states:
\[ \text{adj}(\text{adj}(A)) = |A|^{n-1} A \]
For our matrix \( A \) of order \( 3 \):
\[ \text{adj}(\text{adj}(A)) = |A|^{3-1} A = |A|^2 A \]

4. **Substituting the Determinant**:
We know \( |A| = 2 \):
\[ \text{adj}(\text{adj}(A)) = 2^2 A = 4A \]

### Final Answer:
Thus, the result is:
\[ \text{adj}(\text{adj}(A)) = 4A \]