Find a matrix of order `2xx2`, whose elements are determined by `a_(ij)=(2i-j)^2`

Options

A. `[(9,4),(1,4)]`
B. `[(0,2),(1,3)]`
C. `[(1,0),(9,4)]`
D. `[(4,1),(0,8)]`

Answer

To find a matrix of order \(2 \times 2\) whose elements are determined by the formula \(a_{ij} = (2i - j)^2\), we will calculate each element of the matrix step by step.

### Step 1: Define the matrix structure
A \(2 \times 2\) matrix has the following structure:
\[ \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \]
where \(i\) represents the row number and \(j\) represents the column number.

### Step 2: Calculate \(a_{11}\)
Using the formula \(a_{ij} = (2i - j)^2\):
- For \(i = 1\) and \(j = 1\):
\[ a_{11} = (2 \cdot 1 - 1)^2 = (2 - 1)^2 = 1^2 = 1 \]

### Step 3: Calculate \(a_{12}\)
- For \(i = 1\) and \(j = 2\):
\[ a_{12} = (2 \cdot 1 - 2)^2 = (2 - 2)^2 = 0^2 = 0 \]

### Step 4: Calculate \(a_{21}\)
- For \(i = 2\) and \(j = 1\):
\[ a_{21} = (2 \cdot 2 - 1)^2 = (4 - 1)^2 = 3^2 = 9 \]

### Step 5: Calculate \(a_{22}\)
- For \(i = 2\) and \(j = 2\):
\[ a_{22} = (2 \cdot 2 - 2)^2 = (4 - 2)^2 = 2^2 = 4 \]

### Step 6: Construct the matrix
Now we can construct the matrix using the calculated elements:
\[ \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 9 & 4 \end{bmatrix} \]

### Final Answer
The matrix of order \(2 \times 2\) is:
\[ \begin{bmatrix} 1 & 0 \\ 9 & 4 \end{bmatrix} \]