The construction of `3xx4` matrix a whose element `a_(ij)` is given by `((i+j)^(2))/2` is

Options

A. `[(2,9//2,8,25),(9,4,5,18),(8,25,18,49)]`
B. `[(2,9//2,25//2),(9//2,5//2,5),(25,18,25)]`
C. `[(2,9//2,8,25//2),(9//2,8,25//2,18),(8,25//2,18,49//2)]`
D. None of these

Answer

To construct a \(3 \times 4\) matrix \(A\) where each element \(a_{ij}\) is given by the formula \(\frac{(i+j)^2}{2}\), we will follow these steps:

### Step 1: Define the Matrix Dimensions
We need to create a matrix \(A\) with 3 rows and 4 columns.

### Step 2: Calculate Each Element Using the Given Formula
We will calculate each element \(a_{ij}\) using the formula \(\frac{(i+j)^2}{2}\).

#### Row 1 (i = 1):
- **Element \(a_{11}\)**:
\[ a_{11} = \frac{(1+1)^2}{2} = \frac{2^2}{2} = \frac{4}{2} = 2 \]
- **Element \(a_{12}\)**:
\[ a_{12} = \frac{(1+2)^2}{2} = \frac{3^2}{2} = \frac{9}{2} = 4.5 \]
- **Element \(a_{13}\)**:
\[ a_{13} = \frac{(1+3)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8 \]
- **Element \(a_{14}\)**:
\[ a_{14} = \frac{(1+4)^2}{2} = \frac{5^2}{2} = \frac{25}{2} = 12.5 \]

#### Row 2 (i = 2):
- **Element \(a_{21}\)**:
\[ a_{21} = \frac{(2+1)^2}{2} = \frac{3^2}{2} = \frac{9}{2} = 4.5 \]
- **Element \(a_{22}\)**:
\[ a_{22} = \frac{(2+2)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8 \]
- **Element \(a_{23}\)**:
\[ a_{23} = \frac{(2+3)^2}{2} = \frac{5^2}{2} = \frac{25}{2} = 12.5 \]
- **Element \(a_{24}\)**:
\[ a_{24} = \frac{(2+4)^2}{2} = \frac{6^2}{2} = \frac{36}{2} = 18 \]

#### Row 3 (i = 3):
- **Element \(a_{31}\)**:
\[ a_{31} = \frac{(3+1)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8 \]
- **Element \(a_{32}\)**:
\[ a_{32} = \frac{(3+2)^2}{2} = \frac{5^2}{2} = \frac{25}{2} = 12.5 \]
- **Element \(a_{33}\)**:
\[ a_{33} = \frac{(3+3)^2}{2} = \frac{6^2}{2} = \frac{36}{2} = 18 \]
- **Element \(a_{34}\)**:
\[ a_{34} = \frac{(3+4)^2}{2} = \frac{7^2}{2} = \frac{49}{2} = 24.5 \]

### Step 3: Construct the Matrix
Now we can construct the matrix \(A\) using the calculated elements:

\[ A = \begin{bmatrix} 2 & 4.5 & 8 & 12.5 \\ 4.5 & 8 & 12.5 & 18 \\ 8 & 12.5 & 18 & 24.5 \end{bmatrix} \]

### Final Matrix
Thus, the final \(3 \times 4\) matrix \(A\) is:

\[ A = \begin{bmatrix} 2 & 4.5 & 8 & 12.5 \\ 4.5 & 8 & 12.5 & 18 \\ 8 & 12.5 & 18 & 24.5 \end{bmatrix} \]