Question

Construct a $2\times 2$ matrix, $A=\left[a_{ij}\right]$, whose elements are given by:
$\left(i\right)$ $a_{ij}=\frac{(i+j{)}^2}{2}$
$\left(ii\right)$ $a_{ij}=\frac{i}{j}$
$\left(iii\right)$ $a_{ij}=\frac{(i+2j{)}^2}{2}$

Answer 1

$\left(i\right):$ Since it is a $2\times 2$ matrix.
Let matrix be $A$
Where $A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$
Now it is given that
$a_{ij}=\frac{(i+j{)}^2}{2}$

$a_{ij}$	$i,j$	$a_{ij}=\frac{(i+j{)}^2}{2}$
$a_{11}$	$1,1$	$a_{11}=\frac{(1+1{)}^2}{2}=\frac{(2{)}^2}{2}=2$
$a_{12}$	$1,2$	$a_{12}=\frac{(1+2{)}^2}{2}=\frac{(3{)}^2}{2}=\frac{9}{2}$
$a_{21}$	$2,1$	$a_{21}=\frac{(2+1{)}^2}{2}=\frac{(3{)}^2}{2}=\frac{9}{2}$
$a_{22}$	$2,2$	$a_{22}=\frac{(2+2{)}^2}{2}=\frac{(4{)}^2}{2}=8$

Hence, the required matrix $A$ is
$A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]=\left[\begin{array}{cc}2& \frac{9}{2}\\ \frac{9}{2}& 8\end{array}\right]$

$\left(ii\right):$ Since it is a $2\times 2$ matrix.
Let matrix be $A$
Where $A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$
Now it is given that $a_{ij}=\frac{i}{j}$

$a_{ij}$	$i,j$	$a_{ij}=\frac{i}{j}$
$a_{11}$	$i=1$, $j=1$	$a_{11}=\frac{1}{1}=1$
$a_{12}$	$i=1$, $j=2$	$a_{12}=\frac{1}{2}$
$a_{21}$	$i=2$, $j=1$	$a_{21}=\frac{2}{1}=2$
$a_{22}$	$i=2$, $j=2$	$a_{22}=\frac{2}{2}=1$

Hence, the required matrix $A$ is
$A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]=\left[\begin{array}{cc}1& \frac{1}{2}\\ 2& 1\end{array}\right]$

$\left(iii\right):$ Since it is a $2\times 2$ matrix.
Let matrix be $A$
Where $A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$
Now it is given that
$a_{ij}=\frac{(i+2j{)}^2}{2}$

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

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