Question

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

Answer 1

We have $\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\end{array}\right]$

Since $a_{ij}=\frac{1}{2}|2i-3j|$

$a_{11}=\frac{1}{2}|2\times 1-3\times 1|=\frac{1}{2}|2-3|=\frac{1}{2}|-1|=\frac{1}{2}$

$a_{12}=\frac{1}{2}|2\times 1-3\times 2|=\frac{1}{2}|2-6|=\frac{1}{2}|-4|=\frac{4}{2}=2$

$a_{13}=\frac{1}{2}|2\times 1-3\times 3|=\frac{1}{2}|2-9|=\frac{1}{2}|-7|=\frac{7}{2}$

$a_{21}=\frac{1}{2}|2\times 2-3\times 1|=\frac{1}{2}|4-3|=\frac{1}{2}\left|1\right|=\frac{1}{2}$

$a_{22}=\frac{1}{2}|2\times 2-3\times 2|=\frac{1}{2}|4-6|=\frac{1}{2}|-2|=\frac{2}{2}=1$

$a_{23}=\frac{1}{2}|2\times 2-3\times 3|=\frac{1}{2}|4-9|=\frac{1}{2}|-5|=\frac{5}{2}$

Hence the required matrix is

$A=\left[\begin{array}{ccc}\frac{1}{2}& 2& \frac{7}{2}\\ \frac{1}{2}& 1& \frac{5}{2}\end{array}\right]$