If A is any square matrix of order `3xx3` such that `|A|=3`, then the value of `|adjA|` is ?

Options

A. 3
B. `(1)/(3)`
C. 9
D. 27

Answer

To find the value of \(|\text{adj} A|\) for a square matrix \(A\) of order \(3 \times 3\) where \(|A| = 3\), we can use the property of determinants related to the adjoint of a matrix.

### Step-by-Step Solution:

1. **Understand the relationship between the determinant of a matrix and its adjoint**:
The determinant of the adjoint of a matrix \(A\) is given by the formula:
\[ |\text{adj} A| = |A|^{n-1} \]
where \(n\) is the order of the matrix \(A\).

2. **Identify the order of the matrix**:
Since \(A\) is a \(3 \times 3\) matrix, we have:
\[ n = 3 \]

3. **Substitute the values into the formula**:
We know that \(|A| = 3\). Now, substituting the values into the formula:
\[ |\text{adj} A| = |A|^{3-1} = |A|^{2} \]

4. **Calculate \(|\text{adj} A|\)**:
Now substituting \(|A| = 3\):
\[ |\text{adj} A| = 3^{2} = 9 \]

5. **Conclusion**:
Therefore, the value of \(|\text{adj} A|\) is:
\[ \boxed{9} \]