Question

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

• A. $\left(\begin{array}{ccc}4& 5& 6\\ 0& 3& 4\\ 0& 0& 2\end{array}\right)$
• B. $\left(\begin{array}{ccc}4& 5& 6\\ 3& 4& 0\\ 2& 0& 0\end{array}\right)$
• C. $\left(\begin{array}{ccc}2& 0& 0\\ 3& 4& 0\\ 4& 5& 6\end{array}\right)$
• D. $\left(\begin{array}{ccc}0& 0& 0\\ 1& 2& 3\\ 4& 5& 0\end{array}\right)$

Answer 1

The correct option is C

$\left(\begin{array}{ccc}2& 0& 0\\ 3& 4& 0\\ 4& 5& 6\end{array}\right)$

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 $i<j$ and $a_{ij}=i+j$ if $i\ge j$.
$a_{11}=2,a_{12}=0,a_{13}=0a_{21}=3,a_{22}=4,a_{23}=0a_{31}=4,a_{32}=5,a_{33}=6$

$A=\left(\begin{array}{ccc}2& 0& 0\\ 3& 4& 6\\ 4& 5& 6\end{array}\right)$ is the required matrix