1. **Problem statement:** Write the matrix $M = (a_{ij})$ of order $2 \times 3$ where: - For $i = j$, $a_{ij} = \frac{1}{2}(i^2 + j) - 3j$ - For $i \neq j$, $a_{ij} = 2(i - j)^2 + 3j$ 2. **Matrix order:** $M$ has 2 rows and 3 columns, so $i = 1,2$ and $j = 1,2,3$. 3. **Calculate each element:** - For $i=1, j=1$ (diagonal): $$a_{11} = \frac{1}{2}(1^2 + 1) - 3 \times 1 = \frac{1}{2}(1 + 1) - 3 = \frac{2}{2} - 3 = 1 - 3 = -2$$ - For $i=1, j=2$ (off-diagonal): $$a_{12} = 2(1 - 2)^2 + 3 \times 2 = 2( -1)^2 + 6 = 2 \times 1 + 6 = 8$$ - For $i=1, j=3$ (off-diagonal): $$a_{13} = 2(1 - 3)^2 + 3 \times 3 = 2( -2)^2 + 9 = 2 \times 4 + 9 = 8 + 9 = 17$$ - For $i=2, j=1$ (off-diagonal): $$a_{21} = 2(2 - 1)^2 + 3 \times 1 = 2(1)^2 + 3 = 2 + 3 = 5$$ - For $i=2, j=2$ (diagonal): $$a_{22} = \frac{1}{2}(2^2 + 2) - 3 \times 2 = \frac{1}{2}(4 + 2) - 6 = \frac{6}{2} - 6 = 3 - 6 = -3$$ - For $i=2, j=3$ (off-diagonal): $$a_{23} = 2(2 - 3)^2 + 3 \times 3 = 2( -1)^2 + 9 = 2 + 9 = 11$$ 4. **Final matrix:** $$M = \begin{pmatrix} -2 & 8 & 17 \\ 5 & -3 & 11 \end{pmatrix}$$ This completes the solution for the first problem.