Suppose `A=a_(ij)_(3xx3),` where `a_(ij) epsilonR` if det (adjA)=25, then |det (A)| equals

Options

A. 5
B. 12.5
C. `5sqrt(5)`
D. `5^(2//3)`

Answer

To solve the problem, we need to find the value of \(|\det(A)|\) given that \(\det(\text{adj} A) = 25\) for a \(3 \times 3\) matrix \(A\).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We know that for any square matrix \(A\) of order \(n\), the relationship between the determinant of the adjoint of \(A\) and the determinant of \(A\) is given by:
\[ \det(\text{adj} A) = (\det A)^{n-1} \]
Here, since \(A\) is a \(3 \times 3\) matrix, \(n = 3\).

2. **Applying the Formula**:
Substitute \(n = 3\) into the formula:
\[ \det(\text{adj} A) = (\det A)^{3-1} = (\det A)^2 \]

3. **Setting Up the Equation**:
We know from the problem statement that:
\[ \det(\text{adj} A) = 25 \]
Therefore, we can set up the equation:
\[ (\det A)^2 = 25 \]

4. **Solving for \(\det A\)**:
To find \(\det A\), we take the square root of both sides:
\[ \det A = \pm 5 \]

5. **Finding the Absolute Value**:
The problem asks for \(|\det A|\):
\[ |\det A| = 5 \]

### Final Answer:
Thus, the value of \(|\det A|\) is \(5\).