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

Answer

To find the value of \(|2 \cdot \text{adj} A|\) where \(A\) is a square matrix of order \(3\) and \(|A| = 9\), we can follow these steps:

### Step 1: Understand the relationship between the determinant of a matrix and its adjugate.
The determinant of the adjugate of a matrix \(A\) can be expressed as:
\[ |\text{adj} A| = |A|^{n-1} \]
where \(n\) is the order of the matrix. For a \(3 \times 3\) matrix, \(n = 3\).

### Step 2: Calculate \(|\text{adj} A|\).
Given that \(|A| = 9\), we can substitute this into the formula:
\[ |\text{adj} A| = |A|^{3-1} = |A|^2 = 9^2 = 81 \]

### Step 3: Use the property of determinants with scalar multiplication.
When a matrix is multiplied by a scalar \(k\), the determinant is scaled by \(k^n\), where \(n\) is the order of the matrix. Therefore, for our case:
\[ |kA| = k^n |A| \]
For \(k = 2\) and \(n = 3\):
\[ |2A| = 2^3 |A| = 8 \cdot 9 = 72 \]

### Step 4: Calculate \(|2 \cdot \text{adj} A|\).
Now we can find \(|2 \cdot \text{adj} A|\):
\[ |2 \cdot \text{adj} A| = 2^3 |\text{adj} A| = 8 \cdot 81 = 648 \]

### Final Answer:
Thus, the value of \(|2 \cdot \text{adj} A|\) is:
\[ \boxed{648} \]