Write the element `a_(12)` of the matrix `A=[a_(ij)]_(2xx2)`, whose elements `a_(ij)` are given by `a_(ij)=e^(2ix)sinjx`.

Answer

To find the element \( a_{12} \) of the matrix \( A = [a_{ij}]_{2 \times 2} \), where the elements \( a_{ij} \) are defined by the formula \( a_{ij} = e^{2ix} \sin(jx) \), we will follow these steps:

### Step-by-Step Solution:

1. **Define the Matrix**:
The matrix \( A \) is of order \( 2 \times 2 \). Therefore, it has four elements: \( a_{11}, a_{12}, a_{21}, a_{22} \).

2. **General Element Formula**:
The elements of the matrix are given by the formula:
\[ a_{ij} = e^{2ix} \sin(jx) \]

3. **Find \( a_{12} \)**:
To find the element \( a_{12} \), we need to substitute \( i = 1 \) and \( j = 2 \) into the general formula:
\[ a_{12} = e^{2(1)x} \sin(2x) \]

4. **Simplify the Expression**:
Substitute \( i = 1 \) into \( e^{2ix} \):
\[ a_{12} = e^{2x} \sin(2x) \]

5. **Final Answer**:
Therefore, the element \( a_{12} \) of the matrix \( A \) is:
\[ a_{12} = e^{2x} \sin(2x) \]