A `2xx2` matrix whose elements `a_(ij)` are given by `a_(ij)=i-j` is

Options

A. `[(0,1),(1,0)]`
B. `[(0,-1),(1,0)]`
C. `[(-1,0),(0,1)]`
D. `[(0,1),(-1,0)]`

Answer

To find the elements of a \(2 \times 2\) matrix defined by \(a_{ij} = i - j\), we will follow these steps:

### Step 1: Define the Matrix
We are given a \(2 \times 2\) matrix \(A\) with elements \(a_{ij}\). The elements are defined by the formula \(a_{ij} = i - j\).

### Step 2: Identify the Indices
For a \(2 \times 2\) matrix, the indices \(i\) and \(j\) can take the following values:
- \(i = 1, 2\) (for the rows)
- \(j = 1, 2\) (for the columns)

### Step 3: Calculate Each Element
Now we will calculate each element of the matrix using the formula \(a_{ij} = i - j\):

1. **Calculate \(a_{11}\)**:
\[ a_{11} = 1 - 1 = 0 \]

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

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

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

### Step 4: Construct the Matrix
Now we can construct the matrix \(A\) using the calculated elements:
\[ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]

### Final Answer
The \(2 \times 2\) matrix whose elements \(a_{ij} = i - j\) is:
\[ A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]

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