Let A be a square matrix of order 3 such that transpose of inverse of A is A itself then |adj (adjA)| is equal to

Options

A. 9
B. 27
C. 4
D. 1

Answer

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

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We are given that \((A^{-1})^T = A\). This implies that \(A\) is an orthogonal matrix. For an orthogonal matrix, the inverse is equal to its transpose, i.e., \(A^{-1} = A^T\).

2. **Finding the Determinant of \(A\)**:
For an orthogonal matrix \(A\), the determinant satisfies:
\[ |A|^2 = |I| = 1 \]
Therefore, \(|A| = \pm 1\).

3. **Calculating the Determinant of the Adjoint of \(A\)**:
The adjoint of a matrix \(A\) is related to its determinant by the formula:
\[ |\text{adj} A| = |A|^{n-1} \]
where \(n\) is the order of the matrix. Since \(A\) is a \(3 \times 3\) matrix, we have:
\[ |\text{adj} A| = |A|^{3-1} = |A|^2 \]
From the previous step, we know \(|A|^2 = 1\). Thus:
\[ |\text{adj} A| = 1 \]

4. **Finding the Determinant of the Adjoint of the Adjoint of \(A\)**:
Now we need to find \(|\text{adj}(\text{adj} A)|\). Using the same formula:
\[ |\text{adj}(\text{adj} A)| = |\text{adj} A|^{n-1} \]
Here, \(n\) is still 3, so:
\[ |\text{adj}(\text{adj} A)| = |\text{adj} A|^{3-1} = |\text{adj} A|^2 \]
Since we found \(|\text{adj} A| = 1\):
\[ |\text{adj}(\text{adj} A)| = 1^2 = 1 \]

### Final Answer:
\[ |\text{adj}(\text{adj} A)| = 1 \]