Question

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

Answer 1

(i) Since it is a $2\times 2$ matrix

it has $2$ rows and $2$ column.

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

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

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

Hence,

the required matrix $A$ ia

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

(ii) Since it is a $2\times 2$ matrix

it has $2$ rows and $2$ column.

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

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

(iii) Since it is a $2\times 2$ matrix

it has $2$ rows and $2$ column.

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

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

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

Hence,

the required matrix $A$ ia

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