If determinant of A = 5 and A is a square matrix of order 3 then find the determinant of adj(A)

Explanation

To find the determinant of the adjoint of a matrix \( A \), we can use the following property of determinants:

\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^{n-1} \]

where \( n \) is the order of the square matrix \( A \).

### Step-by-Step Solution:

1. **Identify the given information:**
- We are given that \( \text{det}(A) = 5 \).
- The matrix \( A \) is of order \( 3 \times 3 \), so \( n = 3 \).

2. **Apply the formula for the determinant of the adjoint:**
- We know that \( \text{det}(\text{adj}(A)) = (\text{det}(A))^{n-1} \).
- Here, \( n - 1 = 3 - 1 = 2 \).

3. **Substitute the values into the formula:**
- Substitute \( \text{det}(A) = 5 \) into the formula:
\[ \text{det}(\text{adj}(A)) = (5)^{2} \]

4. **Calculate the result:**
- Now calculate \( 5^{2} \):
\[ \text{det}(\text{adj}(A)) = 25 \]

### Final Answer:
Thus, the determinant of the adjoint of \( A \) is \( 25 \).

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