Construct a 2 × 2 matrix,


Construct a $2 \times 2$ matrix, $A=\left[a_{i j}\right]$, whose elements are given by:

(i) $a_{i j}=\frac{(i+j)^{2}}{2}$

(ii) $a_{i j}=\frac{i}{j}$

(iii) $a_{i j}=\frac{(i+2 j)^{2}}{2}$


In general, a $2 \times 2$ matrix is given by $A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]$

(i) $a_{i j}=\frac{(i+j)^{2}}{2} ; i, j=1,2$

$\therefore a_{11}=\frac{(1+1)^{2}}{2}=\frac{4}{2}=2$    $a_{12}=\frac{(1+2)^{2}}{2}=\frac{9}{2}$

$a_{21}=\frac{(2+1)^{2}}{2}=\frac{9}{2}$    $a_{22}=\frac{(2+2)^{2}}{2}=\frac{16}{2}=8$

Therefore, the required matrix is $A=\left[\begin{array}{ll}2 & \frac{9}{2} \\ \frac{9}{2} & 8\end{array}\right]$

(ii) $a_{i j}=\frac{i}{j}, i, j=1,2$

$\therefore a_{11}=\frac{1}{1}=1 \quad a_{12}=\frac{1}{2}$

$a_{21}=\frac{2}{1}=2 \quad a_{22}=\frac{2}{2}=1$

Therefore, the required matrix is $A=\left[\begin{array}{ll}1 & \frac{1}{2} \\ 2 & 1\end{array}\right]$

(iii) $a_{i j}=\frac{(i+2 j)^{2}}{2}, i, j=1,2$

$\therefore a_{11}=\frac{(1+2)^{2}}{2}=\frac{3^{2}}{2}=\frac{9}{2} \quad a_{12}=\frac{(1+4)^{2}}{2}=\frac{5^{2}}{2}=\frac{25}{2}$

$a_{21}=\frac{(2+2)^{2}}{2}=\frac{4^{2}}{2}=8 \quad a_{22}=\frac{(2+4)^{2}}{2}=\frac{6^{2}}{2}=18$

Therefore, the required matrix is $A=\left[\begin{array}{lr}\frac{9}{2} & \frac{25}{2} \\ 8 & 18\end{array}\right]$


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