Construst a `2xx2` matrix ,A=[`a_(ij)`], whose elements are given by :
`(i) a_(ij)=((i+j)^(2))/(2)(ii)a_(ij)=(i)/(j)`
`(iii) a_(ij)=((i+2j)^(2))/(2)`

Explanation

To construct the \(2 \times 2\) matrices as per the given conditions, we will follow the definitions provided for each element \(a_{ij}\).

### Step-by-step Solution:

**1. Matrix for \(a_{ij} = \frac{(i + j)^2}{2}\)**

- For \(i = 1, j = 1\):
\[ a_{11} = \frac{(1 + 1)^2}{2} = \frac{2^2}{2} = \frac{4}{2} = 2 \]

- For \(i = 1, j = 2\):
\[ a_{12} = \frac{(1 + 2)^2}{2} = \frac{3^2}{2} = \frac{9}{2} = 4.5 \]

- For \(i = 2, j = 1\):
\[ a_{21} = \frac{(2 + 1)^2}{2} = \frac{3^2}{2} = \frac{9}{2} = 4.5 \]

- For \(i = 2, j = 2\):
\[ a_{22} = \frac{(2 + 2)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8 \]

Thus, the first matrix \(A_1\) is:
\[ A_1 = \begin{bmatrix} 2 & 4.5 \\ 4.5 & 8 \end{bmatrix} \]

**2. Matrix for \(a_{ij} = \frac{i}{j}\)**

- For \(i = 1, j = 1\):
\[ a_{11} = \frac{1}{1} = 1 \]

- For \(i = 1, j = 2\):
\[ a_{12} = \frac{1}{2} = 0.5 \]

- For \(i = 2, j = 1\):
\[ a_{21} = \frac{2}{1} = 2 \]

- For \(i = 2, j = 2\):
\[ a_{22} = \frac{2}{2} = 1 \]

Thus, the second matrix \(A_2\) is:
\[ A_2 = \begin{bmatrix} 1 & 0.5 \\ 2 & 1 \end{bmatrix} \]

**3. Matrix for \(a_{ij} = \frac{(i + 2j)^2}{2}\)**

- For \(i = 1, j = 1\):
\[ a_{11} = \frac{(1 + 2 \cdot 1)^2}{2} = \frac{(1 + 2)^2}{2} = \frac{3^2}{2} = \frac{9}{2} = 4.5 \]

- For \(i = 1, j = 2\):
\[ a_{12} = \frac{(1 + 2 \cdot 2)^2}{2} = \frac{(1 + 4)^2}{2} = \frac{5^2}{2} = \frac{25}{2} = 12.5 \]

- For \(i = 2, j = 1\):
\[ a_{21} = \frac{(2 + 2 \cdot 1)^2}{2} = \frac{(2 + 2)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8 \]

- For \(i = 2, j = 2\):
\[ a_{22} = \frac{(2 + 2 \cdot 2)^2}{2} = \frac{(2 + 4)^2}{2} = \frac{6^2}{2} = \frac{36}{2} = 18 \]

Thus, the third matrix \(A_3\) is:
\[ A_3 = \begin{bmatrix} 4.5 & 12.5 \\ 8 & 18 \end{bmatrix} \]

### Summary of Matrices:
1. \(A_1 = \begin{bmatrix} 2 & 4.5 \\ 4.5 & 8 \end{bmatrix}\)
2. \(A_2 = \begin{bmatrix} 1 & 0.5 \\ 2 & 1 \end{bmatrix}\)
3. \(A_3 = \begin{bmatrix} 4.5 & 12.5 \\ 8 & 18 \end{bmatrix}\)