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Answer:  $A= \begin{bmatrix} e^{2x}\sin x &e^{2x}\sin 2x \\ e^{4x} \sin x& e^{4x}\sin 2x \end{bmatrix}$

Given:  $a_{ij}= e^{2x}\sin xj$
Here we have to construct $2\times 2$  matrix according to  $e^{2x}\sin xj$

Hint:  Putting the value of each row and column element according to the question in matrix

Solution:  Let  $A= \left [ a_{ij} \right ]_{2\times 2}$  $= e^{2x}\sin xj$

So, the elements in a  $2\times 2$  are $a_{11},a_{12},a_{21},a_{22}$

$A= \begin{bmatrix} a_{11} &a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

$\! \! \! \! \! \! \! \! \! a_{11}= e^{2\times 1x}\sin x\times 1= e^{2x}\sin x\\a_{12}= e^{2\times 1x}\sin x\times 2= e^{2x}\sin 2x$           $\! \! \! \! \! \! \! \! \! a_{21}= e^{2\times 2x}\sin x\times 1= e^{4x}\sin x\\a_{22}= e^{2\times 2x}\sin x\times 2= e^{4x}\sin 2x$

Substituting these values in Matrix $A$ , we get

$A= \begin{bmatrix} e^{2x}\sin x &e^{2x}\sin 2x \\ e^{4x} \sin x& e^{4x}\sin 2x \end{bmatrix}$

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