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

Since, $A = {\left[ {{a_{ij}}} \right]_{m \times n}}1 \le i \le m$ and $1 \le j \le n,i,j \in N$

$\therefore$ $A = {\left[ {{e^{i \cdot x}}\sin jx} \right]_{3 \times 2}};1 \le i \le 3;1 \le j \le 2$

$\Rightarrow$ ${a_{11}} = {e^{1 \cdot x}} \cdot \sin 1 \cdot x = {e^x}\sin x$

${a_{12}} = {e^{1 \cdot x}} \cdot \sin 2 \cdot x = {e^x}\sin 2x$

${a_{21}} = {e^{2 \cdot x}} \cdot \sin 1 \cdot x = {e^{2x}}\sin x$
${a_{22}} = {e^{2 \cdot x}} \cdot \sin 2 \cdot x = {e^{2x}}\sin 2x$

${a_{31}} = {e^{3 \cdot x}} \cdot \sin 1 \cdot x = {e^{3x}}\sin x$
${a_{32}} = {e^{3 \cdot x}} \cdot \sin 2 \cdot x = {e^{3x}}\sin 2x$
$\therefore$ $A = {\left[ {\begin{array}{cccccccccccccccccccc}{{e^x}\sin x}&{{e^x}\sin 2x}\\{{e^{2x}}\sin x}&{{e^{2x}}\sin 2x}\\{{e^{3x}}\sin x}&{{e^{3x}}\sin 2x}\end{array}} \right]_{3 \times 2}}$