1. Construct the 2x2 matrices: (a) $a_{ij} = \frac{(i - 2j)^2}{2}$ for $i,j=1,2$: Calculate each element: $a_{11} = \frac{(1-2\cdot1)^2}{2} = \frac{(-1)^2}{2} = \frac{1}{2}$ $a_{12} = \frac{(1-2\cdot2)^2}{2} = \frac{(-3)^2}{2} = \frac{9}{2}$ $a_{21} = \frac{(2-2\cdot1)^2}{2} = \frac{0^2}{2} = 0$ $a_{22} = \frac{(2-2\cdot2)^2}{2} = \frac{(-2)^2}{2} = 2$ Matrix: $$\begin{bmatrix}\frac{1}{2} & \frac{9}{2}\\ 0 & 2\end{bmatrix}$$ (b) $a_{ij} = \frac{(i+2j)^2}{2}$: $a_{11} = \frac{(1+2\cdot1)^2}{2} = \frac{3^2}{2} = \frac{9}{2}$ $a_{12} = \frac{(1+2\cdot2)^2}{2} = \frac{5^2}{2} = \frac{25}{2}$ $a_{21} = \frac{(2+2\cdot1)^2}{2} = \frac{4^2}{2} = 8$ $a_{22} = \frac{(2+2\cdot2)^2}{2} = \frac{6^2}{2} = 18$ Matrix: $$\begin{bmatrix}\frac{9}{2} & \frac{25}{2}\\ 8 & 18\end{bmatrix}$$ (c) $a_{ij} = -2i + 3j$: $a_{11} = -2\cdot1 + 3\cdot1 = 1$ $a_{12} = -2\cdot1 + 3\cdot2 = 4$ $a_{21} = -2\cdot2 + 3\cdot1 = -1$ $a_{22} = -2\cdot2 + 3\cdot2 = 2$ Matrix: $$\begin{bmatrix}1 & 4\\ -1 & 2\end{bmatrix}$$ 2. Construct a 3x2 matrix: (a) $a_{ij} = \frac{1}{2} |i - 3j|$ for $i=1,2,3$ and $j=1,2$: Calculate elements: $i=1,j=1: \frac{1}{2}|1-3|=\frac{1}{2}*2=1$ $i=1,j=2: \frac{1}{2}|1-6|=\frac{1}{2}*5=2.5$ $i=2,j=1: \frac{1}{2}|2-3|=\frac{1}{2}*1=0.5$ $i=2,j=2: \frac{1}{2}|2-6|=\frac{1}{2}*4=2$ $i=3,j=1: \frac{1}{2}|3-3|=0$ $i=3,j=2: \frac{1}{2}|3-6|=\frac{1}{2}*3=1.5$ Matrix: $$\begin{bmatrix}1 & 2.5\\ 0.5 & 2\\ 0 & 1.5\end{bmatrix}$$ (b) For matrix $A=[a_{ij}]$ with $a_{ij} = \begin{cases} i+j, & i \geq j \\ i - j, & i < j \end{cases}$ For $3 \times 2$ matrix $i=1..3$, $j=1..2$: Calculate elements: $i=1,j=1$: $1+1=2$ (since $i\geq j$) $i=1,j=2$: $1-2=-1$ (since $i