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 a \(2 \times 2\) matrix \(A = [a_{ij}]\) based on the given formulas, we will evaluate each element of the matrix according to the specified rules.

### Step 1: Define the Matrix Structure
The matrix \(A\) will have the following structure:
\[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \]

### Step 2: Calculate Elements Using the First Formula
The first formula is given by:
\[ a_{ij} = \frac{(i + j)^2}{2} \]

- **Calculate \(a_{11}\)**:
\[ a_{11} = \frac{(1 + 1)^2}{2} = \frac{2^2}{2} = \frac{4}{2} = 2 \]

- **Calculate \(a_{12}\)**:
\[ a_{12} = \frac{(1 + 2)^2}{2} = \frac{3^2}{2} = \frac{9}{2} \]

- **Calculate \(a_{21}\)**:
\[ a_{21} = \frac{(2 + 1)^2}{2} = \frac{3^2}{2} = \frac{9}{2} \]

- **Calculate \(a_{22}\)**:
\[ a_{22} = \frac{(2 + 2)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8 \]

Now, substituting these values into the matrix \(A\):
\[ A = \begin{bmatrix} 2 & \frac{9}{2} \\ \frac{9}{2} & 8 \end{bmatrix} \]

### Step 3: Calculate Elements Using the Second Formula
The second formula is given by:
\[ a_{ij} = \frac{i}{j} \]

- **Calculate \(a_{11}\)**:
\[ a_{11} = \frac{1}{1} = 1 \]

- **Calculate \(a_{12}\)**:
\[ a_{12} = \frac{1}{2} = \frac{1}{2} \]

- **Calculate \(a_{21}\)**:
\[ a_{21} = \frac{2}{1} = 2 \]

- **Calculate \(a_{22}\)**:
\[ a_{22} = \frac{2}{2} = 1 \]

Now, substituting these values into the matrix \(A\):
\[ A = \begin{bmatrix} 1 & \frac{1}{2} \\ 2 & 1 \end{bmatrix} \]

### Step 4: Calculate Elements Using the Third Formula
The third formula is given by:
\[ a_{ij} = \frac{(i + 2j)^2}{2} \]

- **Calculate \(a_{11}\)**:
\[ a_{11} = \frac{(1 + 2 \cdot 1)^2}{2} = \frac{(1 + 2)^2}{2} = \frac{3^2}{2} = \frac{9}{2} \]

- **Calculate \(a_{12}\)**:
\[ a_{12} = \frac{(1 + 2 \cdot 2)^2}{2} = \frac{(1 + 4)^2}{2} = \frac{5^2}{2} = \frac{25}{2} \]

- **Calculate \(a_{21}\)**:
\[ a_{21} = \frac{(2 + 2 \cdot 1)^2}{2} = \frac{(2 + 2)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8 \]

- **Calculate \(a_{22}\)**:
\[ a_{22} = \frac{(2 + 2 \cdot 2)^2}{2} = \frac{(2 + 4)^2}{2} = \frac{6^2}{2} = \frac{36}{2} = 18 \]

Now, substituting these values into the matrix \(A\):
\[ A = \begin{bmatrix} \frac{9}{2} & \frac{25}{2} \\ 8 & 18 \end{bmatrix} \]

### Summary of the Matrices
1. From the first formula:
\[ A = \begin{bmatrix} 2 & \frac{9}{2} \\ \frac{9}{2} & 8 \end{bmatrix} \]

2. From the second formula:
\[ A = \begin{bmatrix} 1 & \frac{1}{2} \\ 2 & 1 \end{bmatrix} \]

3. From the third formula:
\[ A = \begin{bmatrix} \frac{9}{2} & \frac{25}{2} \\ 8 & 18 \end{bmatrix} \]