If A is a 3 x 3 matrix such that |5-adjA|=5 , then |A| is equal to :

Options

A. `pm1/5`
B. `pm5`
C. `pm1`
D. `pm1/25`

Answer

To solve the problem, we start with the given information about the matrix \( A \) and its adjoint.

### Step-by-step Solution:

1. **Understanding the Given Condition**:
We are given that \( |5 \cdot \text{adj} A| = 5 \).

2. **Using the Property of Determinants**:
Recall the property of determinants: if \( k \) is a constant and \( A \) is an \( n \times n \) matrix, then:
\[ |kA| = k^n |A| \]
In our case, \( k = 5 \) and the order \( n = 3 \) (since \( A \) is a \( 3 \times 3 \) matrix). Therefore:
\[ |5 \cdot \text{adj} A| = 5^3 |\text{adj} A| = 125 |\text{adj} A| \]

3. **Setting Up the Equation**:
From the given condition, we have:
\[ 125 |\text{adj} A| = 5 \]
Dividing both sides by 125, we find:
\[ |\text{adj} A| = \frac{5}{125} = \frac{1}{25} \]

4. **Using the Relationship Between the Adjoint and the Determinant**:
The determinant of the adjoint of a matrix \( A \) is related to the determinant of \( A \) by the formula:
\[ |\text{adj} A| = |A|^{n-1} \]
For a \( 3 \times 3 \) matrix, \( n = 3 \), so:
\[ |\text{adj} A| = |A|^{3-1} = |A|^2 \]
Substituting the value we found:
\[ |A|^2 = \frac{1}{25} \]

5. **Solving for \( |A| \)**:
Taking the square root of both sides gives us:
\[ |A| = \pm \sqrt{\frac{1}{25}} = \pm \frac{1}{5} \]

### Final Answer:
Thus, the value of \( |A| \) is:
\[ |A| = \pm \frac{1}{5} \]