Let A be the the square matrix of order 3 and deteminant of A is 5 then find the value of determinant of adj(A)

Options

A. 7I
B. 5I
C. 3I
D. 25

Answer

To find the value of the determinant of the adjoint of matrix \( A \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Order of the Matrix**:
Since \( A \) is a square matrix of order 3, we have \( n = 3 \).

2. **Use the Formula for Determinant of Adjoint**:
The determinant of the adjoint of a matrix \( A \) is given by the formula:
\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^{n-1} \]
where \( n \) is the order of the matrix.

3. **Substitute the Given Values**:
We know that \( \text{det}(A) = 5 \) and \( n = 3 \). Therefore:
\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^{3-1} = (\text{det}(A))^{2} \]

4. **Calculate**:
Now, substituting the value of \( \text{det}(A) \):
\[ \text{det}(\text{adj}(A)) = (5)^{2} = 25 \]

5. **Final Answer**:
Thus, the value of the determinant of the adjoint of \( A \) is:
\[ \text{det}(\text{adj}(A)) = 25 \]