If `A=[a_(ij)]_(2xx2)`, where `a_(ij)=(i+2j)^2/2`, then A is equal to:

Options

A. `[(9,25),(8,18)]`
B. `[(9//2,25//2),(8,18)]`
C. `[(9,25),(4,9)]`
D. `[(9//2,25//2),(4,9)]`

Answer

To find the matrix \( A = [a_{ij}]_{2 \times 2} \) where \( a_{ij} = \frac{(i + 2j)^2}{2} \), we will calculate each element of the matrix step by step.

### Step 1: Identify the elements of the matrix
The matrix \( A \) is a \( 2 \times 2 \) matrix, which means it has four elements \( a_{11}, a_{12}, a_{21}, a_{22} \).

### Step 2: Calculate \( a_{11} \)
For \( i = 1 \) and \( j = 1 \):
\[ a_{11} = \frac{(1 + 2 \cdot 1)^2}{2} = \frac{(1 + 2)^2}{2} = \frac{3^2}{2} = \frac{9}{2} \]

### Step 3: Calculate \( a_{12} \)
For \( i = 1 \) and \( j = 2 \):
\[ a_{12} = \frac{(1 + 2 \cdot 2)^2}{2} = \frac{(1 + 4)^2}{2} = \frac{5^2}{2} = \frac{25}{2} \]

### Step 4: Calculate \( a_{21} \)
For \( i = 2 \) and \( j = 1 \):
\[ a_{21} = \frac{(2 + 2 \cdot 1)^2}{2} = \frac{(2 + 2)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8 \]

### Step 5: Calculate \( a_{22} \)
For \( i = 2 \) and \( j = 2 \):
\[ a_{22} = \frac{(2 + 2 \cdot 2)^2}{2} = \frac{(2 + 4)^2}{2} = \frac{6^2}{2} = \frac{36}{2} = 18 \]

### Step 6: Form the matrix \( A \)
Now that we have all the elements, we can write the matrix \( A \):
\[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} \frac{9}{2} & \frac{25}{2} \\ 8 & 18 \end{bmatrix} \]

### Final Answer
Thus, the matrix \( A \) is:
\[ A = \begin{bmatrix} \frac{9}{2} & \frac{25}{2} \\ 8 & 18 \end{bmatrix} \]