Question

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

$a_{11}=\left[\frac{1}{1}\right]=\left[1\right]=1$

$a_{12}=\left[\frac{1}{2}\right]=\left[0.5\right]=0$

$a_{13}=\left[\frac{1}{3}\right]=\left[0.33\right]=0$

$a_{21}=\left[\frac{2}{1}\right]=\left[2\right]=2$

$a_{22}=\left[\frac{2}{2}\right]=\left[1\right]=1$

$a_{23}=\left[\frac{2}{3}\right]=\left[0.67\right]=0$

Hence the required matrix is

$A=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\end{array}\right]$