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 value of \(|\text{adj}(\text{adj}(\text{adj} A))|\) given that \(|A| = 2\) and \(A\) is a \(3 \times 3\) matrix.

### Step-by-Step Solution:

1. **Understanding the Determinant of the Adjoint**:
The determinant of the adjoint of a matrix \(A\) of order \(n\) is given by the formula:
\[ |\text{adj}(A)| = |A|^{n-1} \]
For our case, since \(A\) is a \(3 \times 3\) matrix, \(n = 3\).

2. **Finding the Determinant of the First Adjoint**:
Using the formula:
\[ |\text{adj}(A)| = |A|^{3-1} = |A|^2 \]
Given \(|A| = 2\), we have:
\[ |\text{adj}(A)| = 2^2 = 4 \]

3. **Finding the Determinant of the Second Adjoint**:
Now, we need to find \(|\text{adj}(\text{adj}(A))|\):
\[ |\text{adj}(\text{adj}(A))| = |\text{adj}(A)|^{3-1} = |\text{adj}(A)|^2 \]
Substituting the value we found:
\[ |\text{adj}(\text{adj}(A))| = 4^2 = 16 \]

4. **Finding the Determinant of the Third Adjoint**:
Finally, we find \(|\text{adj}(\text{adj}(\text{adj}(A)))|\):
\[ |\text{adj}(\text{adj}(\text{adj}(A)))| = |\text{adj}(\text{adj}(A))|^{3-1} = |\text{adj}(\text{adj}(A))|^2 \]
Substituting the value we found:
\[ |\text{adj}(\text{adj}(\text{adj}(A)))| = 16^2 = 256 \]

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