A ` 2 xx 1 ` matrix whose elements are given by `a _(ij) = |i-j|^(2)` is :

Options

A. `[{:(1,0):}]`
B. `[{:(0),(1):}]`
C. `[{:(0,1):}]`
D. `[{:(1),(0):}]`

Answer

To solve the problem, we need to construct a \(2 \times 1\) matrix \(A\) using the given formula for its elements, which is \(a_{ij} = |i - j|^2\).

### Step-by-Step Solution:

1. **Identify the size of the matrix**:
We are given that the matrix is \(2 \times 1\). This means it has 2 rows and 1 column. We can denote the elements of the matrix as:
\[ A = \begin{pmatrix} A_{11} \\ A_{21} \end{pmatrix} \]

2. **Calculate the first element \(A_{11}\)**:
Using the formula \(a_{ij} = |i - j|^2\), we substitute \(i = 1\) and \(j = 1\):
\[ A_{11} = |1 - 1|^2 = |0|^2 = 0 \]

3. **Calculate the second element \(A_{21}\)**:
Now, we substitute \(i = 2\) and \(j = 1\):
\[ A_{21} = |2 - 1|^2 = |1|^2 = 1 \]

4. **Construct the matrix**:
Now that we have both elements, we can write the matrix \(A\):
\[ A = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \]

5. **Conclusion**:
The \(2 \times 1\) matrix \(A\) whose elements are given by \(a_{ij} = |i - j|^2\) is:
\[ A = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \]