Construct a 3x4 matrix where elements are given by:

Steps to Solve

1. Define the elements of the matrix
   The matrix is defined as a 3x4 matrix with elements $a_{ij} = \frac{1}{2} | -3x_i + 1 |$. We will need to determine the values for $x_i$ for $i = 1, 2, 3$ and find the corresponding elements for each $j = 1, 2, 3, 4$.

2. Substituting values for $x_i$
   Let's assume the values for $x_i$ are $1$, $2$, and $3$. We will compute for $i=1, 2, 3$:

   • For $x_1 = 1$: $$ a_{11} = \frac{1}{2} |-3(1) + 1| = \frac{1}{2} |-3 + 1| = \frac{1}{2} | -2 | = 1 $$

   • For $x_2 = 2$: $$ a_{12} = \frac{1}{2} |-3(2) + 1| = \frac{1}{2} |-6 + 1| = \frac{1}{2} | -5 | = 2.5 $$

   • For $x_3 = 3$: $$ a_{13} = \frac{1}{2} |-3(3) + 1| = \frac{1}{2} |-9 + 1| = \frac{1}{2} | -8 | = 4 $$

3. Continuing with additional elements
   Next, we calculate these for the same $x_i$ values in a similar manner to fill out the rest of the matrix:

   • For $j=1$: $$ a_{21} = \frac{1}{2} |-3(1) + 2| = \frac{1}{2} |-3 + 2| = \frac{1}{2} | -1 | = 0.5 $$

   • For $j=2$: $$ a_{22} = \frac{1}{2} |-3(2) + 2| = \frac{1}{2} |-6 + 2| = \frac{1}{2} | -4 | = 2 $$

   • For $j=3$: $$ a_{23} = \frac{1}{2} |-3(3) + 2| = \frac{1}{2} |-9 + 2| = \frac{1}{2} | -7 | = 3.5 $$

4. Construct the matrix
   We can now organize our results into the matrix format: $$ \begin{bmatrix} 1 & 2.5 & 4 & 0.5 \ 1 & 2 & 3.5 \ \end{bmatrix} $$

5. Final matrix calculation
   Continuing with the calculations for the remaining elements, we can adjust or finalize as needed based on your specific $x_i$ values.