If A and B are square matrices of same order `3` , such that `|A|=2` and `AB=2I`, write the value of `|B|`

Answer

To solve the problem, we need to find the value of \(|B|\) given that \(|A| = 2\) and \(AB = 2I\), where \(A\) and \(B\) are square matrices of order 3.

### Step-by-step Solution:

1. **Given Information**:
- We know that \(|A| = 2\).
- We also know that \(AB = 2I\).

2. **Taking Determinants**:
- We can take the determinant of both sides of the equation \(AB = 2I\).
- This gives us \(|AB| = |2I|\).

3. **Using the Property of Determinants**:
- The property of determinants states that \(|AB| = |A| \cdot |B|\).
- Therefore, we have \(|A| \cdot |B| = |2I|\).

4. **Calculating \(|2I|\)**:
- The determinant of a scalar multiple of the identity matrix can be calculated as follows:
\[ |cI| = c^n \]
where \(c\) is a scalar and \(n\) is the order of the matrix.
- Here, \(c = 2\) and \(n = 3\), so:
\[ |2I| = 2^3 = 8. \]

5. **Substituting the Values**:
- Now we can substitute the values we have into the equation:
\[ |A| \cdot |B| = 8. \]
- We know \(|A| = 2\), so:
\[ 2 \cdot |B| = 8. \]

6. **Solving for \(|B|\)**:
- To find \(|B|\), we divide both sides of the equation by 2:
\[ |B| = \frac{8}{2} = 4. \]

### Final Answer:
\[ |B| = 4. \]