If `A` is a square matrix of order 3 such that `|A|=2,t h e n|(a d jA^(-1))^(-1)|` is ___________.

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**:
We know that for any square matrix \(A\), the determinant of the adjoint of \(A\) is given by:
\[ |adj A| = |A|^{n-1} \]
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. **Finding the Determinant of \(A^{-1}\)**:
The determinant of the inverse of a matrix is given by:
\[ |A^{-1}| = \frac{1}{|A|} \]
Given \(|A| = 2\), we have:
\[ |A^{-1}| = \frac{1}{2} \]

3. **Finding the Determinant of the Adjoint of \(A^{-1}\)**:
Now, we can find the determinant of the adjoint of \(A^{-1}\):
\[ |adj A^{-1}| = |A^{-1}|^{3-1} = |A^{-1}|^2 \]
Substituting \(|A^{-1}| = \frac{1}{2}\):
\[ |adj A^{-1}| = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

4. **Finding the Determinant of \((adj A^{-1})^{-1}\)**:
The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix. Therefore:
\[ |(adj A^{-1})^{-1}| = \frac{1}{|adj A^{-1}|} = \frac{1}{\frac{1}{4}} = 4 \]

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

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