construct a matrix of order 3×4 if aij=(i-j)^2/2

To construct a matrix \( A \) of order \( 3 \times 4 \) with elements defined by the formula \( a_{ij} = \frac{(i-j)^2}{2} \), we need to calculate the elements for each \( i \) (row) and \( j \) (column). Here's how we can compute the elements:

### Step-by-Step Calculation:
- For \( i = 1 \) and \( j = 1 \):  
  \( a_{11} = \frac{(1-1)^2}{2} = \frac{0^2}{2} = 0 \)

- For \( i = 1 \) and \( j = 2 \):  
  \( a_{12} = \frac{(1-2)^2}{2} = \frac{(-1)^2}{2} = \frac{1}{2} = 0.5 \)

- For \( i = 1 \) and \( j = 3 \):  
  \( a_{13} = \frac{(1-3)^2}{2} = \frac{(-2)^2}{2} = \frac{4}{2} = 2 \)

- For \( i = 1 \) and \( j = 4 \):  
  \( a_{14} = \frac{(1-4)^2}{2} = \frac{(-3)^2}{2} = \frac{9}{2} = 4.5 \)

We continue this process for each value of \( i \) and \( j \) to fill out the matrix.

### Matrix \( A \) (3×4):
\[
A = \begin{pmatrix}
0 & 0.5 & 2 & 4.5 \\
0.5 & 0 & 0.5 & 2 \\
2 & 0.5 & 0 & 0.5 \\
\end{pmatrix}
\]

This is the required matrix of order \( 3 \times 4 \).

Would you like more details or have any questions? Here are 5 related questions:

1. How would you construct a similar matrix for different dimensions, say 4×5?
2. What are the properties of matrices created with such a formula?
3. How do you determine the rank of this matrix?
4. Can this matrix be simplified or transformed using elementary row operations?
5. How does the matrix change if the formula is modified to \( a_{ij} = (i+j)^2/2 \)?

**Tip:** When working with matrices, always check for symmetry and other special properties based on the formula given for the elements.

Learn how to construct a 3×4 matrix using the formula a_{ij} = (i-j)^2/2 with a detailed step-by-step calculation. This problem covers matrix algebra concepts and computation of matrix elements based on a specific formula.

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