Construct a `2xx2` matrix `A=[a_(ij)]` whose elements are given by :
`a_(ij)=(2i-j)/(3)`

Answer

To construct a \(2 \times 2\) matrix \(A = [a_{ij}]\) whose elements are defined by the formula \(a_{ij} = \frac{2i - j}{3}\), we will follow these steps:

### Step 1: Identify the indices \(i\) and \(j\)
Since we are constructing a \(2 \times 2\) matrix, the indices \(i\) and \(j\) can take the values:
- \(i = 1, 2\)
- \(j = 1, 2\)

### Step 2: Calculate the elements of the matrix
We will calculate each element of the matrix using the given formula \(a_{ij} = \frac{2i - j}{3}\).

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

2. **Calculate \(a_{12}\)**:
\[ a_{12} = \frac{2 \cdot 1 - 2}{3} = \frac{2 - 2}{3} = \frac{0}{3} = 0 \]

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

4. **Calculate \(a_{22}\)**:
\[ a_{22} = \frac{2 \cdot 2 - 2}{3} = \frac{4 - 2}{3} = \frac{2}{3} \]

### Step 3: Construct the matrix
Now that we have all the elements, we can construct the matrix \(A\):
\[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} \frac{1}{3} & 0 \\ 1 & \frac{2}{3} \end{bmatrix} \]

### Final Result
The constructed matrix \(A\) is:
\[ A = \begin{bmatrix} \frac{1}{3} & 0 \\ 1 & \frac{2}{3} \end{bmatrix} \]

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