Question

Construct a 3 × 4 matrix, whose elements are given by ${\displaystyle a_{ij}=2i-j}$

Answer 1

In general, a 3 × 4 matrix is given by ${\displaystyle A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& a_{14}\\ a_{21}& a_{22}& a_{23}& a_{24}\\ a_{31}& a_{32}& a_{33}& a_{34}\end{array}\right]}$

⇒ ${\displaystyle a_{ij}=2i-j,i=1,2,3\text{and}j=1,2,3,4}$

${\displaystyle \therefore a_{11}=2\times 1-1=2-1=1}$

${\displaystyle a_{21}=2\times 2-1=4-1=3}$

${\displaystyle a_{31}=2\times 3-1=6-1=5}$

⇒ ${\displaystyle a_{12}=2\times 1-2=2-2=0}$

${\displaystyle a_{22}=2\times 2-2=4-2=2}$

${\displaystyle a_{32}=2\times 3-2=6-2=4}$

⇒ ${\displaystyle a_{13}=2\times 1-3=2-3=-1}$ 

${\displaystyle a_{23}=2\times 2-3=4-3=1}$

${\displaystyle a_{33}=2\times 3-3=6-3=3}$

⇒ ${\displaystyle a_{14}=2\times 1-4=2-4=-2}$

${\displaystyle a_{24}=2\times 2-4=4-4=0}$

${\displaystyle a_{34}=2\times 3-4=6-4=2}$

Therefore, the required matrix is ${\displaystyle A=\left[\begin{array}{cccc}1& 0& -1& -2\\ 3& 2& 1& 0\\ 5& 4& 3& 2\end{array}\right]}$