Question

Construct a $2\times 3$ matrix $A=\left[a_{ij}\right]$ whose elements are given by $a_{ij}=\left\{\frac{2i}{3j}\right\}$, where $\{.\}$ denotes the fractional part function.

Answer 1

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

Since $a_{ij}=\left\{\frac{2i}{3j}\right\}$

$a_{11}=\left\{\frac{2\times 1}{3\times 1}\right\}=\left\{\frac{2}{3}\right\}=\frac{2}{3}$

$a_{12}=\left\{\frac{2\times 1}{3\times 2}\right\}=\left\{\frac{1}{3}\right\}=\frac{1}{3}$

$a_{13}=\left\{\frac{2\times 1}{3\times 3}\right\}=\left\{\frac{2}{9}\right\}=\frac{2}{9}$

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

$a_{22}=\left\{\frac{2\times 2}{3\times 2}\right\}=\left\{\frac{2}{3}\right\}=\frac{2}{3}$

$a_{23}=\left\{\frac{2\times 2}{3\times 3}\right\}=\left\{\frac{4}{9}\right\}=\frac{4}{9}$

Hence the required matrix is

$A=\left[\begin{array}{ccc}\frac{2}{3}& \frac{1}{3}& \frac{2}{9}\\ \frac{1}{3}& \frac{2}{3}& \frac{4}{9}\end{array}\right]$