Construct a matrix `[a_(ij)]_(3xx3)`,where `a_(ij)=(i-j)/(i+j).`

Explanation

To construct the matrix \( A = [a_{ij}]_{3 \times 3} \) where \( a_{ij} = \frac{i - j}{i + j} \), we will follow these steps:

### Step 1: Define the matrix structure
We need to create a \( 3 \times 3 \) matrix, which means it will have 3 rows and 3 columns. The elements of the matrix are denoted as \( a_{ij} \), where \( i \) represents the row number and \( j \) represents the column number.

### Step 2: Calculate each element of the matrix
We will calculate each element \( a_{ij} \) using the formula \( a_{ij} = \frac{i - j}{i + j} \).

#### Row 1:
1. **Calculate \( a_{11} \)**:
\[ a_{11} = \frac{1 - 1}{1 + 1} = \frac{0}{2} = 0 \]

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

3. **Calculate \( a_{13} \)**:
\[ a_{13} = \frac{1 - 3}{1 + 3} = \frac{-2}{4} = -\frac{1}{2} \]

#### Row 2:
4. **Calculate \( a_{21} \)**:
\[ a_{21} = \frac{2 - 1}{2 + 1} = \frac{1}{3} \]

5. **Calculate \( a_{22} \)**:
\[ a_{22} = \frac{2 - 2}{2 + 2} = \frac{0}{4} = 0 \]

6. **Calculate \( a_{23} \)**:
\[ a_{23} = \frac{2 - 3}{2 + 3} = \frac{-1}{5} \]

#### Row 3:
7. **Calculate \( a_{31} \)**:
\[ a_{31} = \frac{3 - 1}{3 + 1} = \frac{2}{4} = \frac{1}{2} \]

8. **Calculate \( a_{32} \)**:
\[ a_{32} = \frac{3 - 2}{3 + 2} = \frac{1}{5} \]

9. **Calculate \( a_{33} \)**:
\[ a_{33} = \frac{3 - 3}{3 + 3} = \frac{0}{6} = 0 \]

### Step 3: Construct the matrix
Now that we have calculated all the elements, we can construct the matrix \( A \):

\[ A = \begin{bmatrix} 0 & -\frac{1}{3} & -\frac{1}{2} \\ \frac{1}{3} & 0 & -\frac{1}{5} \\ \frac{1}{2} & \frac{1}{5} & 0 \end{bmatrix} \]

### Summary of the matrix:
The final matrix \( A \) is:
\[ A = \begin{bmatrix} 0 & -\frac{1}{3} & -\frac{1}{2} \\ \frac{1}{3} & 0 & -\frac{1}{5} \\ \frac{1}{2} & \frac{1}{5} & 0 \end{bmatrix} \]