Question

Construct a $\mathbf{2}\mathbf{\times }\mathbf{2}$ matrix, whose elements ${\mathit{a}}_{\mathbf{i}\mathbf{j}}$ are given, ${{\mathbf{a}}_{\mathbf{i}\mathbf{j}}\mathbf{=}\mathbf{\left(}\frac{\mathbf{i}\mathbf{-}\mathbf{j}}{\mathbf{i}\mathbf{\pm }\mathbf{j}}\mathbf{\right)}}^{\mathbf{2}}$.

Answer 1

The matrix $\mathit{A}$ of order $\mathbf{2}\mathbf{\times }\mathbf{2}$:
$\mathrm{A}={\left[a_{ij}\right]}_{2\times 2}=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$

The entries of the matrix A:

Substitute $i=1,2$ and $j=1,2$ in the given $a_{ij}$.
$a_{11}={\left(\frac{1-1}{1+1}\right)}^2={\left(\frac{0}{2}\right)}^2=0$
$a_{12}={\left(\frac{1-2}{1+2}\right)}^2={\left(\frac{-1}{3}\right)}^2=\frac{1}{9}$
$a_{21}={\left(\frac{2-1}{2+1}\right)}^2={\left(\frac{1}{3}\right)}^2=\frac{1}{9}$
$a_{22}={\left(\frac{2-2}{2+2}\right)}^2={\left(\frac{0}{4}\right)}^2=0$

Hence, $\mathbf{2}\mathbf{\times }\mathbf{2}$ matrix $\mathrm{A}=\left[\begin{array}{cc}0& \frac{1}{9}\\ \frac{1}{9}& 0\end{array}\right]$