If A is order 3 square matrix such that `|A|=2`, then `|"adj (adj (adj A))"|` is

Options

A. 512
B. 256
C. 64
D. none of these

Explanation

To solve the problem, we need to find the determinant of the adjoint of the adjoint of the adjoint of a 3x3 matrix \( A \) given that \( |A| = 2 \).

### Step-by-Step Solution:

1. **Understanding the Determinant of the Adjoint**:
The determinant of the adjoint of a matrix \( A \) can be expressed as:
\[ |\text{adj}(A)| = |A|^{n-1} \]
where \( n \) is the order of the matrix. For a 3x3 matrix, \( n = 3 \).

2. **Calculating the Determinant of the First Adjoint**:
Given \( |A| = 2 \), we can find the determinant of the first adjoint:
\[ |\text{adj}(A)| = |A|^{3-1} = |A|^2 = 2^2 = 4 \]

3. **Calculating the Determinant of the Second Adjoint**:
Now, we find the determinant of the adjoint of the adjoint:
\[ |\text{adj}(\text{adj}(A))| = |\text{adj}(A)|^{3-1} = |\text{adj}(A)|^2 = 4^2 = 16 \]

4. **Calculating the Determinant of the Third Adjoint**:
Finally, we calculate the determinant of the adjoint of the adjoint of the adjoint:
\[ |\text{adj}(\text{adj}(\text{adj}(A)))| = |\text{adj}(\text{adj}(A))|^{3-1} = |\text{adj}(\text{adj}(A))|^2 = 16^2 = 256 \]

5. **Final Result**:
Therefore, the determinant of the adjoint of the adjoint of the adjoint of \( A \) is:
\[ |\text{adj}(\text{adj}(\text{adj}(A)))| = 256 \]

### Answer:
The final answer is \( 256 \).