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# Construct a $3\times 2$ matrix whose elements are given by $a_{ij}=e^{i.x}\sin jx$

$\begin{array}{1 1} A=\begin{bmatrix}sin x& e^x sin2x\\e^{2x}sin x& e^{2x} sin2x\\e^{3x}sin x& e^{3x} sin2x\end{bmatrix} \\ A=\begin{bmatrix}e^xsin x& e^x \\e^{2x}sin x& e^{2x} sin2x\\e^{3x}sin x& e^{3x} sin2x\end{bmatrix} \\ A=\begin{bmatrix}e^xsin x& e^x sin2x\\e^{2x}sin x& e^{2x} sin2x\\e^{3x}sin x& e^{3x} sin2x\end{bmatrix} \\A=\begin{bmatrix}e^xsin x& e^x sin2x\\e^{2x}sin x& e^{2x} sin2x\\e^{3x}sin x& e^{3} sin2x\end{bmatrix} \end{array}$

Toolbox:
• In general $a_{3\times 2}$ matrix is given by $A=\begin{bmatrix}a_{11} & a_{12}\\a_{21} &a_{22}\\a_{31} & a_{32}\end{bmatrix}$
Step1;
Given:
$a_{ij}=e^{i.x}sin jx$ where replace i=1,2,3 and j=1,2
Hence $a_{11}=e^{1.x}sin 1x$(Replacing i=1,j=1)
$\;\;\;\;\;\;\;\;\;\;\;\;\;=e^xsin x$
$a_{12}=e^{i.x}sin jx$(Replacing i=1,j=2)
$\;\;\;\;\;\;=e^xsin 2x$
Step2:
$a_{21}=e^{i.x}sin jx$(Replacing i=2,j=1)
$\;\;\;\;\;\;=e^{2x}sin x$
$a_{22}=e^{i.x}sin jx$(Replacing i=2,j=1)
$\;\;\;\;\;\;=e^{2x}sin 2x$
Step3:
$a_{31}=e^{i.x}sin jx$(Replacing i=3,j=1)
$\;\;\;\;\;\;=e^{3x}sin x$
$a_{32}=e^{i.x}sin jx$(Replacing i=3,j=2)
$\;\;\;\;\;\;=e^{3x}sin 2x$
Step4:
Hence we have :-
$A=\begin{bmatrix}e^xsin x& e^x sin2x\\e^{2x}sin x& e^{2x} sin2x\\e^{3x}sin x& e^{3x} sin2x\end{bmatrix}$