Constructing a Matrix using aij = e2ix sin(xj)
Question:
Construct a \( 2 \times 2 \) matrix \( A = [a_{ij}] \) whose elements are given by \( a_{ij} = e^{2ix}\sin(xj) \).
Step 1: Matrix Order
- Rows → \( i = 1, 2 \)
- Columns → \( j = 1, 2 \)
Step 2: Compute Elements
For \( i = 1 \):
\[ a_{11} = e^{2x}\sin(x),\quad a_{12} = e^{2x}\sin(2x) \]
For \( i = 2 \):
\[ a_{21} = e^{4x}\sin(x),\quad a_{22} = e^{4x}\sin(2x) \]
Step 3: Form the Matrix
\[ A = \begin{bmatrix} e^{2x}\sin x & e^{2x}\sin 2x \\ e^{4x}\sin x & e^{4x}\sin 2x \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} e^{2x}\sin x & e^{2x}\sin 2x \\ e^{4x}\sin x & e^{4x}\sin 2x \end{bmatrix} \]