If `A` is a square matrix of order `3` such that `|A|=2`, then `|(adjA^(-1))^(-1)|` is

Options

A. `1`
B. `2`
C. `4`
D. `8`

Explanation

To solve the problem, we need to find the value of \( |(adj A^{-1})^{-1}| \) given that \( |A| = 2 \) and \( A \) is a square matrix of order 3.

### Step-by-Step Solution:

1. **Understanding the Determinant of the Adjoint**:
The determinant of the adjoint of a matrix \( A \) can be expressed as:
\[ |adj A| = |A|^{n-1} \]
where \( n \) is the order of the matrix. Since \( A \) is a \( 3 \times 3 \) matrix, \( n = 3 \). Therefore:
\[ |adj A| = |A|^{3-1} = |A|^2 \]

2. **Calculating the Determinant of the Adjoint of the Inverse**:
We know that:
\[ |A| = 2 \]
Thus:
\[ |adj A| = |A|^2 = 2^2 = 4 \]

3. **Finding the Determinant of the Inverse**:
The determinant of the inverse of a matrix is given by:
\[ |A^{-1}| = \frac{1}{|A|} \]
Therefore:
\[ |A^{-1}| = \frac{1}{2} \]

4. **Finding the Determinant of the Adjoint of the Inverse**:
We apply the formula for the adjoint again:
\[ |adj A^{-1}| = |A^{-1}|^{n-1} = |A^{-1}|^{3-1} = |A^{-1}|^2 \]
Substituting the value we found:
\[ |adj A^{-1}| = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

5. **Finding the Determinant of the Inverse of the Adjoint**:
Finally, we need to find \( |(adj A^{-1})^{-1}| \):
\[ |(adj A^{-1})^{-1}| = \frac{1}{|adj A^{-1}|} \]
Therefore:
\[ |(adj A^{-1})^{-1}| = \frac{1}{\frac{1}{4}} = 4 \]

### Final Answer:
Thus, the value of \( |(adj A^{-1})^{-1}| \) is \( 4 \).