Construct a 4 × 3 matrix whose element are: a_ij= i-j i+j - Vidyadip

Question

Construct a 4 × 3 matrix whose element are:
$\text{a}_\text{ij}=\frac{\text{i}-\text{j}}{\text{i}+\text{j}}$

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Answer

Here,
$\text{a}_{11}=\frac{1-1}{1+1}=\frac{0}{2},\text{ a}_{12}=\frac{1-2}{1+2}=\frac{-1}{3},$ $\text{a}_{13}=\frac{1-3}{1+3}=\frac{-2}{4}=\frac{-1}{2}$
$\text{a}_{21}=\frac{2-1}{2+1}=\frac{1}{3},\text{ a}_{22}=\frac{2-2}{2-2}=\frac{0}{0}=0,$ $\text{a}_{23}=\frac{2-3}{2+3}=\frac{-1}{5}$
$\text{a}_{31}=\frac{3-1}{3+1}=\frac{2}{4}=\frac{1}{2},\text{ a}_{32}=\frac{3-2}{3+2}=\frac{1}{5},$ $\text{a}_{33}=\frac{3-3}{3+3}=\frac{0}{6}=0$
$\text{a}_{41}=\frac{4-1}{4+1}=\frac{3}{5},\text{a}_{42}=\frac{4-2}{4+2}=\frac{2}{6}=\frac{1}{3}$ and $\text{a}_{43}=\frac{4-3}{4+3}=\frac{1}{7}$
So, the required matrix is $\begin{bmatrix}0&\frac{-1}{3}&\frac{-1}{2}\\\frac {1}{3}&0&\frac{-1}{5}\\\frac{1}{2}&\frac{1}{5}&0\\ \frac{3}{5}&\frac{1}{3}&\frac{1}{7}\end{bmatrix}.$

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