Construct a 2 x 2 matrix A=[aij], such that aij = (i + j)^2 / 2.

Steps to Solve

1. Matrix Size Definition We need to construct a $2 \times 2$ matrix A, so we will have 2 rows and 2 columns.

2. Element Calculation The elements of the matrix are defined as $a_{ij} = \frac{(i + j)^2}{2}$. We will calculate each element based on the indices $i$ and $j$.

3. Calculate (a_{11}) For (i = 1), (j = 1): $$a_{11} = \frac{(1 + 1)^2}{2} = \frac{2^2}{2} = \frac{4}{2} = 2$$

4. Calculate (a_{12}) For (i = 1), (j = 2): $$a_{12} = \frac{(1 + 2)^2}{2} = \frac{3^2}{2} = \frac{9}{2} = 4.5$$

5. Calculate (a_{21}) For (i = 2), (j = 1): $$a_{21} = \frac{(2 + 1)^2}{2} = \frac{3^2}{2} = \frac{9}{2} = 4.5$$

6. Calculate (a_{22}) For (i = 2), (j = 2): $$a_{22} = \frac{(2 + 2)^2}{2} = \frac{4^2}{2} = \frac{16}{2} = 8$$

7. Matrix Construction Now we can construct the matrix A using the computed elements: $$ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} $$ Which gives: $$ A = \begin{bmatrix} 2 & 4.5 \ 4.5 & 8 \end{bmatrix} $$