Question

Construct a $3\times 2$ matrix whose elements are given by $a_{ij}=\frac{1}{2}|i-3j|.$

Answer 1

Let the matrix be $A$.
Since it is $3\times 2$ matrix,
$A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right]$
Now it is given that $a_{ij}=\frac{1}{2}|i-3j|$

$i,j$	$a_{ij}=\frac{1}{2}|i-3j|$
$1,1$	$a_{11}=\frac{1}{2}|1-3\times 1|=\frac{1}{2}|-2|=1$
$2,1$	$a_{21}=\frac{1}{2}|2-3\times 1|=\frac{1}{2}|-1|=\frac{1}{2}$
$3,1$	$a_{31}=\frac{1}{2}|3-3\times 1|=\frac{1}{2}|3-3|=0$
$1,2$	$a_{12}=\frac{1}{2}|1-3\times 2|=\frac{1}{2}|-5|=\frac{5}{2}$
$2,2$	$a_{22}=\frac{1}{2}|2-3\times 2|=\frac{1}{2}|-4|=2$
$3,2$	$a_{32}=\frac{1}{2}|3-3\times 2|=\frac{1}{2}|-3|=\frac{3}{2}$

Hence the required matrix is

$A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\\ a_{21}& a_{32}\end{array}\right]=\left[\begin{array}{cc}1& \frac{5}{2}\\ \frac{1}{2}& 2\\ 0& \frac{3}{2}\end{array}\right]$