Question

Construct a $3\times 3$ matrix whose elements $a_{ij}$ are given by $a_{ij}=0$ if $i\ne j$ and $a_{ij}=i\times j$ if $i=j$

Answer 1

The elements of a 3\times 3 matrix are: $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$

Applying the given rule: $a_{ij}=0$ if $i\ne$j and $a_{ij}=i\times j$, if $i=j$

$a_{11}=1,a_{12}=0,a_{13}=0a_{21}=0,a_{22}=4,a_{23}=0a_{31}=0,a_{32}=0,a_{33}=9$

Therefore $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 4& 0\\ 0& 0& 9\end{array}\right]$ is the required matrix.