Construct a 2*2 matrix 𝐴 = [𝑎𝑖𝑗] whose element is given by 𝑎𝑖𝑗 = {
(𝑖−𝑗
2
2
}

To construct a 2×2 matrix \( A = [a_{ij}] \) where the element \( a_{ij} \) is defined by the formula:

\[
a_{ij} = \frac{(i - j)^2}{2}
\]

Let's calculate each element:

### Step-by-step calculation:

- For \( a_{11} \): 
\[
a_{11} = \frac{(1 - 1)^2}{2} = \frac{0^2}{2} = 0
\]

- For \( a_{12} \): 
\[
a_{12} = \frac{(1 - 2)^2}{2} = \frac{(-1)^2}{2} = \frac{1}{2}
\]

- For \( a_{21} \): 
\[
a_{21} = \frac{(2 - 1)^2}{2} = \frac{1^2}{2} = \frac{1}{2}
\]

- For \( a_{22} \): 
\[
a_{22} = \frac{(2 - 2)^2}{2} = \frac{0^2}{2} = 0
\]

### Final matrix \( A \):

\[
A = \begin{bmatrix} 
0 & \frac{1}{2} \\ 
\frac{1}{2} & 0 
\end{bmatrix}
\]

Would you like further details or explanations?

### Related questions:
1. How would the matrix change if we expanded to a 3×3 matrix?
2. What is the determinant of the matrix \( A \)?
3. Can this matrix be diagonalized, and if so, how?
4. What are the eigenvalues and eigenvectors of matrix \( A \)?
5. How would the formula \( \frac{(i - j)^2}{2} \) affect an \( n \times n \) matrix?

### Tip:
When working with symmetric matrices like this, the off-diagonal elements play a key role in determining its properties, such as eigenvalues and diagonalizability.

Construct a 2×2 matrix A = [a_{ij}] whose element is given by a_{ij} = (i−j)^2 / 2

Learn how to construct a 2×2 symmetric matrix using the formula (i−j)^2 / 2. This problem involves key linear algebra concepts and step-by-step matrix element calculations. Ideal for students in Grades 10-12.

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