A matrix of order `2 xx 2` whose elements `a_("ij")` are given by `a_("ij") = (1)/(2)|(i + j)|^(2)`, is a:

Options

A. symmetric matrix
B. skew-symmetric matrix
C. rectangular matrix
D. singular matrix

Answer

To solve the problem, we need to find the elements of a \(2 \times 2\) matrix defined by the formula \(a_{ij} = \frac{1}{2} |i + j|^2\). Let's break it down step by step.

### Step 1: Identify the elements of the matrix

The matrix \(A\) has the following elements:
- \(a_{11}\)
- \(a_{12}\)
- \(a_{21}\)
- \(a_{22}\)

### Step 2: Calculate each element using the given formula

1. **Calculate \(a_{11}\)**:
\[ a_{11} = \frac{1}{2} |1 + 1|^2 = \frac{1}{2} |2|^2 = \frac{1}{2} \cdot 4 = 2 \]

2. **Calculate \(a_{12}\)**:
\[ a_{12} = \frac{1}{2} |1 + 2|^2 = \frac{1}{2} |3|^2 = \frac{1}{2} \cdot 9 = 4.5 \]

3. **Calculate \(a_{21}\)**:
\[ a_{21} = \frac{1}{2} |2 + 1|^2 = \frac{1}{2} |3|^2 = \frac{1}{2} \cdot 9 = 4.5 \]

4. **Calculate \(a_{22}\)**:
\[ a_{22} = \frac{1}{2} |2 + 2|^2 = \frac{1}{2} |4|^2 = \frac{1}{2} \cdot 16 = 8 \]

### Step 3: Construct the matrix

Now that we have all the elements, we can construct the matrix \(A\):
\[ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} 2 & 4.5 \\ 4.5 & 8 \end{pmatrix} \]

### Step 4: Determine the properties of the matrix

1. **Check if the matrix is singular**:
A matrix is singular if its determinant is zero. The determinant of a \(2 \times 2\) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by \(ad - bc\).
\[ \text{Determinant} = (2)(8) - (4.5)(4.5) = 16 - 20.25 = -4.25 \neq 0 \]
Thus, the matrix is not singular.

2. **Check if the matrix is symmetric**:
A matrix is symmetric if \(A = A^T\). The transpose of matrix \(A\) is:
\[ A^T = \begin{pmatrix} 2 & 4.5 \\ 4.5 & 8 \end{pmatrix} \]
Since \(A = A^T\), the matrix is symmetric.

### Conclusion

The matrix \(A\) is symmetric and not singular.