construct a 2 2 matrix A = [a_ ij ] whose elemants a_ ij are give… - Vidyadip

Question

construct a $2\times 2$ matrix $A = [a_{ij}]$ whose elemants $a_{ij}$ are given by $a_{ij} =\left\{\begin{array}{cl}\frac{|-3\text{i}+\text{j}|}{2} & ,\text{if i }\neq\text{j} \\ (\text{i}+\text{j})^2 & , \text{if i }=\text{j}\end{array}\right.$

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Answer

Let us the matrix be $\text{A}=\begin{bmatrix}\text{a}_{11}&\text{a}_{12}\\\text{a}_{21}&\text{a}_{22} \end{bmatrix}$
For the entries $a_{12}$ and $a_{21 }$ we have $i \neq j,$ so by the
Condition we have $\text{a}_{12}=\frac{|-3\text{i}+\text{j}|}{2},\text{ a}_{21}=\frac{|-3\text{i}+\text{j}|}{2}$
$\Rightarrow\text{a}_{12}=\frac{|-3+2|}{2},\text{ a}_{21}=\frac{|-3.2+1|}{2}$
$\Rightarrow\text{a}_{12}=\frac{|-1|}{2},\text{ a}_{21}=\frac{|-5|}{2}$
$\Rightarrow\text{a}_{12}=\frac{1}{2},\text{ a}_{21}=\frac{5}{2}$
For the entries $a_{11}$ and $a_{22}$ we have $i = j,$ so by the given condition we have
$\Rightarrow\text{a}_{11}=(\text{i + j})^2, \text{ a}_{22}=(\text{i + j})^2$
$\Rightarrow\text{a}_{11}=(1+1)^2,\text{ a}_{22}=(2+2)^2$
$\Rightarrow\text{a}_{11}=4,\text{ a}_{22}=16$
So, $\text{A}=\begin{bmatrix}4&\frac{1}{2}\\\frac{5}{2}&16 \end{bmatrix}$

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