Construct a matrix `A=[a_(ij)]_(2xx2)` whose elements `a_(ij)` are given by `a_(ij)=e^(2ix) sin jx`.

Answer

To construct a matrix \( A = [a_{ij}]_{2 \times 2} \) whose elements are defined by \( a_{ij} = e^{2ix} \sin(jx) \), we will follow these steps:

### Step 1: Define the Matrix
We need to create a \( 2 \times 2 \) matrix, which means we will have four elements: \( a_{11}, a_{12}, a_{21}, a_{22} \).

### Step 2: Calculate Each Element
- **Element \( a_{11} \)**:
\[ a_{11} = e^{2i \cdot x} \sin(1 \cdot x) = e^{2ix} \sin(x) \]

- **Element \( a_{12} \)**:
\[ a_{12} = e^{2i \cdot x} \sin(2 \cdot x) = e^{2ix} \sin(2x) \]

- **Element \( a_{21} \)**:
\[ a_{21} = e^{2 \cdot 2i \cdot x} \sin(1 \cdot x) = e^{4ix} \sin(x) \]

- **Element \( a_{22} \)**:
\[ a_{22} = e^{2 \cdot 2i \cdot x} \sin(2 \cdot x) = e^{4ix} \sin(2x) \]

### Step 3: Construct the Matrix
Now we can write the matrix \( A \) using the calculated elements:
\[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} e^{2ix} \sin(x) & e^{2ix} \sin(2x) \\ e^{4ix} \sin(x) & e^{4ix} \sin(2x) \end{bmatrix} \]

### Final Matrix
Thus, the final matrix \( A \) is:
\[ A = \begin{bmatrix} e^{2ix} \sin(x) & e^{2ix} \sin(2x) \\ e^{4ix} \sin(x) & e^{4ix} \sin(2x) \end{bmatrix} \]

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