If A is a `3xx3` such that `|5.adj(A)|=5` then `|A|` is equal to

Options

A. `+-1/5`
B. `+-1/25`
C. `+-1`
D. `+-5`

Explanation

To solve the problem, we need to find the value of \(|A|\) given that \(|5 \cdot \text{adj}(A)| = 5\) for a \(3 \times 3\) matrix \(A\).

### Step-by-Step Solution:

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

2. **Using the Determinant Property**:
The property of determinants states that \(|k \cdot B| = k^n \cdot |B|\), where \(k\) is a scalar and \(B\) is a matrix of size \(n \times n\). Here, \(n = 3\) (since \(A\) is a \(3 \times 3\) matrix).

Thus, we can write:
\[ |5 \cdot \text{adj}(A)| = 5^3 \cdot |\text{adj}(A)| \]
This simplifies to:
\[ |5 \cdot \text{adj}(A)| = 125 \cdot |\text{adj}(A)| \]

3. **Setting Up the Equation**:
From the problem statement, we have:
\[ 125 \cdot |\text{adj}(A)| = 5 \]
To find \(|\text{adj}(A)|\), we divide both sides by 125:
\[ |\text{adj}(A)| = \frac{5}{125} = \frac{1}{25} \]

4. **Using the Determinant of the Adjoint**:
There is a property that states:
\[ |\text{adj}(A)| = |A|^{n-1} \]
For a \(3 \times 3\) matrix, \(n = 3\), so:
\[ |\text{adj}(A)| = |A|^{3-1} = |A|^2 \]
Therefore, we can substitute:
\[ |A|^2 = \frac{1}{25} \]

5. **Finding the Determinant of A**:
To find \(|A|\), we take the square root of both sides:
\[ |A| = \pm \sqrt{\frac{1}{25}} = \pm \frac{1}{5} \]

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