Question

Construct a $2\times 2$ matrix $A=\left[a_{ij}\right]$, whose element $a_{ij}$ is $a_{ij}=\frac{(2i+j{)}^2}{2}$.

Answer 1

$a_{ij}=\frac{(2i+j{)}^2}{2}$

$a_{ij}$	$i=,j=$	$a_{ij}=\frac{(2i+j{)}^2}{2}$
$a_{11}$	$i=1,j=1$	$a_{11}=\frac{\left(2\right(1)+1{)}^2}{2}=\frac{(3{)}^2}{2}=\frac{9}{2}$
$a_{12}$	$i=1,j=2$	$a_{12}=\frac{\left(2\right(1)+2{)}^2}{2}=\frac{(4{)}^2}{2}=\frac{16}{2}=8$
$a_{21}$	$i=2,j=1$	$a_{21}=\frac{\left(2\right(2)+1{)}^2}{2}=\frac{(5{)}^2}{2}=\frac{25}{2}$
$a_{22}$	$i=2,j=2$	$a_{22}=\frac{\left(2\right(2)+2{)}^2}{2}=\frac{(6{)}^2}{2}=\frac{36}{2}=18$

$\Rightarrow$ Required matrix $A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]=\left[\begin{array}{cc}\frac{9}{2}& 8\\ \frac{25}{2}& 18\end{array}\right]$