If A is a sqaure matrix of order `3xx3` and `|A|=5`, then `|adj.A|` is

Options

A. `5`
B. `125`
C. `15`
D. `25`

Answer

To find the determinant of the adjoint of a matrix \( A \) where \( A \) is a square matrix of order \( 3 \times 3 \) and \( |A| = 5 \), we can use the property of determinants related to the adjoint of a matrix.

### Step-by-Step Solution:

1. **Understanding the property of the adjoint**:
The determinant of the adjoint of a matrix \( A \) (denoted as \( |adj(A)| \)) is given by the formula:
\[ |adj(A)| = |A|^{n-1} \]
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. **Substituting the values into the formula**:
Now, substituting \( n = 3 \) into the formula:
\[ |adj(A)| = |A|^{3-1} = |A|^{2} \]

4. **Using the given determinant of \( A \)**:
We know that \( |A| = 5 \). Therefore:
\[ |adj(A)| = 5^{2} \]

5. **Calculating the result**:
Now, calculating \( 5^{2} \):
\[ |adj(A)| = 25 \]

Thus, the determinant of the adjoint of matrix \( A \) is \( 25 \).

### Final Answer:
\[ |adj(A)| = 25 \]