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

Answer

To find the value of \(|B|\) given that \(|A| = 2\) and \(AB = 2I\), we can follow these steps:

### Step 1: Use the property of determinants
We know that for any two square matrices \(A\) and \(B\) of the same order, the determinant of their product is the product of their determinants. Thus, we have:
\[ |AB| = |A| \cdot |B| \]

### Step 2: Calculate the determinant of \(AB\)
Given that \(AB = 2I\), we can find the determinant of \(AB\):
\[ |AB| = |2I| \]
The determinant of a scalar multiple of the identity matrix \(kI\) (where \(k\) is a scalar and \(I\) is the identity matrix) is given by:
\[ |kI| = k^n \quad \text{(where \(n\) is the order of the matrix)} \]
In this case, since \(I\) is a \(3 \times 3\) matrix, we have:
\[ |2I| = 2^3 = 8 \]

### Step 3: Set up the equation
Now we can substitute this back into our equation from Step 1:
\[ |A| \cdot |B| = |AB| \]
Substituting the known values:
\[ 2 \cdot |B| = 8 \]

### Step 4: Solve for \(|B|\)
Now, we can solve for \(|B|\):
\[ |B| = \frac{8}{2} = 4 \]

### Conclusion
Thus, the value of \(|B|\) is:
\[ \boxed{4} \]