Construct a 3 3 matrix whose elements are given by a_ i j =( i + … - Vidyadip

Question

Construct a $3 \times 3$ matrix whose elements are given by $a_{i j}=\frac{( i + j )^3}{3}$

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Answer

$\begin{aligned} & a_{11}=\frac{(1+1)^3}{3}=\frac{2^3}{3}=\frac{8}{3} \\ & a_{12}=\frac{(1+2)^3}{3}=\frac{27}{3}=9 \\ & a_{13}=\frac{(1+3)^3}{3}=\frac{64}{3}=\frac{64}{3} \\ & a_{21}=\frac{(2+1)^3}{3}=\frac{27}{3}=9 \\ & a_{22}=\frac{(2+2)^3}{3}=\frac{64}{3}=\frac{64}{3} \\ & a_{23}=\frac{(2+3)^3}{3}=\frac{125}{3}=\frac{125}{3} \\ & a_{31}=\frac{(3+1)^3}{3}=\frac{64}{3}=\frac{64}{3} \\ & a_{32}=\frac{(3+2)^3}{3}=\frac{125}{3}=\frac{125}{3} \\ & a_{33}=\frac{(3+3)^3}{3}=\frac{216}{3}=72\end{aligned}$
The required matrix $A=\left[\begin{array}{ccc}\frac{8}{3} & 9 & \frac{64}{3} \\ 9 & \frac{64}{3} & \frac{125}{3} \\ \frac{64}{3} & \frac{125}{3} & 72\end{array}\right]$

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