If A is a square matrix of order `3` with `|A|=9`, then write the value of `|2.adj.A|`.

Answer

To find the value of \(|2 \cdot \text{adj}(A)|\), we can use the properties of determinants and adjugates. Let's go through the solution step by step.

Given:
- \( A \) is a square matrix of order 3.
- \(|A| = 9\).

We need to find \(|2 \cdot \text{adj}(A)|\).

### Step-by-Step Solution:

1. **Recall the property of determinants with scalar multiplication:**
For any scalar \( k \) and square matrix \( A \) of order \( n \),
\[ |kA| = k^n \cdot |A| \]
Here, \( k = 2 \) and the order \( n = 3 \).

\[ |2A| = 2^3 \cdot |A| = 8 \cdot |A| \]

2. **Recall the property of the determinant of the adjugate matrix:**
For any square matrix \( A \) of order \( n \),
\[ |\text{adj}(A)| = |A|^{n-1} \]
Here, \( n = 3 \).

\[ |\text{adj}(A)| = |A|^{3-1} = |A|^2 \]

3. **Calculate \(|\text{adj}(A)|\) using the given \(|A| = 9\):**
\[ |\text{adj}(A)| = 9^2 = 81 \]

4. **Combine the results to find \(|2 \cdot \text{adj}(A)|\):**
Using the property of determinants with scalar multiplication again, where \( k = 2 \) and the matrix is \(\text{adj}(A)\) of order 3,
\[ |2 \cdot \text{adj}(A)| = 2^3 \cdot |\text{adj}(A)| = 8 \cdot 81 \]

5. **Calculate the final value:**
\[ 8 \cdot 81 = 648 \]

Therefore, the value of \(|2 \cdot \text{adj}(A)|\) is \( 648 \).