Question

Construct a 2 × 3 matrix whose elements a_ij are given by :
(i) a_ij = i . j
(ii) a_ij = 2i − j
(iii) a_ij = i + j
(iv) a_ij = $\frac{(i+j{)}^2}{2}$

Answer 1

$\left(i\right)$

Here,
$a_{ij}=i.j$, $1\le i\le 2$ and $1\le j\le 3$

$$\begin{gathered} a_{11}=1\times 1=1,a_{12}=1\times 2=2,a_{13}=1\times 3=3 \\ {a}_{21}=2\times 1=2,a_{22}=2\times 2=4\mathrm{and}{a}_{23}=2\times 3=6 \end{gathered}$$

Required matrix = A = $\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\end{array}\right]$

$\left(ii\right)$

Here,
$a_{ij}=2i-j$

$$\begin{gathered} a_{11}=2\left(1\right)-1=2-1=1,a_{12}=2\left(1\right)-2=2-2=0,a_{13}=2\left(1\right)-3=2-3=-1 \\ {a}_{21}=2\left(2\right)-1=4-1=3,a_{22}=2\left(2\right)-2=4-2=2\mathrm{and}{a}_{23}=2\left(2\right)-3=4-3=1 \end{gathered}$$

Required matrix = A = $\left[\begin{array}{ccc}1& 0& -1\\ 3& 2& 1\end{array}\right]$

$\left(iii\right)$
Here,
$a_{ij}=i+j$

$$\begin{gathered} a_{11}=1+1=2,a_{12}=1+2=3,a_{13}=1+3=4 \\ {a}_{21}=2+1=3,a_{22}=2+2=4\mathrm{and}{a}_{23}=2+3=5 \end{gathered}$$

Required matrix = A = $\left[\begin{array}{ccc}2& 3& 4\\ 3& 4& 5\end{array}\right]$

$\left(iv\right)$

Here,
$a_{ij}=\frac{{\left(i+j\right)}^2}{2}$

$$\begin{gathered} a_{11}=\frac{{\left(1+1\right)}^2}{2}=\frac{{\left(2\right)}^2}{2}=\frac{4}{2}=2,a_{12}=\frac{{\left(1+2\right)}^2}{2}=\frac{{\left(3\right)}^2}{2}=\frac{9}{2},a_{13}=\frac{{\left(1+3\right)}^2}{2}=\frac{{\left(4\right)}^2}{2}=\frac{16}{2}=8 \\ {a}_{21}=\frac{{\left(2+1\right)}^2}{2}=\frac{{\left(3\right)}^2}{2}=\frac{9}{2},a_{22}=\frac{{\left(2+2\right)}^2}{2}=\frac{{\left(4\right)}^2}{2}=\frac{16}{2}=8\mathrm{and}{a}_{23}=\frac{{\left(2+3\right)}^2}{2}=\frac{{\left(5\right)}^2}{2}=\frac{25}{2} \end{gathered}$$

Required matrix = A = $\left[\begin{array}{ccc}2& \frac{9}{2}& 8\\ \frac{9}{2}& 8& \frac{25}{2}\end{array}\right]$