construct a 2 2 matrix A = [ a _ ij ] whose elements a _ ij are g… - Vidyadip

Question

construct a $2 \times 2$ matrix $A = [{a}_{ij}]$ whose elements ${a}_{ij}$ are given by ${a}_{ij}$ $=\begin{cases}\frac{|-3i+j|}{2},&\text{if i }\neq\text{j}\\(i+j)^2,&\text{if i }=\text{j}\end{cases}$

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Answer

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

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