7. Peter, John and Mary intend to purchase \( 15,16,18 \) books and \( 9,11,15 \) pens each respectively. In shop A, a book cost KShs 250 and a pens costs KShs 175 while in shop B, a book cost KShs 225 while a pen costs KShs 190 . i. Represent this information in matrix form: ii. Using matrix method, determine the total cost the three students would sy to acquire the items from shop A and from shop B. (4 mar)

To represent the information in matrix form and calculate the total cost for Peter, John, and Mary to acquire the items from both shops, we can follow these steps: ### i. Representing the Information in Matrix Form Let: - \( B \) be the matrix representing the number of books each student intends to buy. - \( P \) be the matrix representing the number of pens each student intends to buy. - \( C_A \) be the cost matrix for shop A. - \( C_B \) be the cost matrix for shop B. 1. **Books Matrix (B)**: \[ B = \begin{bmatrix} 15 \\ 16 \\ 18 \end{bmatrix} \] 2. **Pens Matrix (P)**: \[ P = \begin{bmatrix} 9 \\ 11 \\ 15 \end{bmatrix} \] 3. **Cost Matrix for Shop A (C_A)**: \[ C_A = \begin{bmatrix} 250 & 175 \end{bmatrix} \] 4. **Cost Matrix for Shop B (C_B)**: \[ C_B = \begin{bmatrix} 225 & 190 \end{bmatrix} \] ### ii. Using Matrix Method to Determine Total Cost To find the total cost for each shop, we will perform matrix multiplication. 1. **Total Cost from Shop A**: The total cost from shop A can be calculated as follows: \[ \text{Total Cost}_A = B^T \cdot C_A^T + P^T \cdot C_A^T \] Where \( B^T \) and \( P^T \) are the transposes of the books and pens matrices. First, we calculate the total cost for books and pens separately: - For books: \[ \text{Total Cost for Books}_A = B^T \cdot \begin{bmatrix} 250 \\ 0 \end{bmatrix} = 15 \cdot 250 + 16 \cdot 250 + 18 \cdot 250 \] \[ = 3750 + 4000 + 4500 = 12250 \] - For pens: \[ \text{Total Cost for Pens}_A = P^T \cdot \begin{bmatrix} 0 \\ 175 \end{bmatrix} = 9 \cdot 175 + 11 \cdot 175 + 15 \cdot 175 \] \[ = 1575 + 1925 + 2625 = 6125 \] - Total Cost from Shop A: \[ \text{Total Cost}_A = 12250 + 6125 = 18375 \] 2. **Total Cost from Shop B**: Similarly, we calculate the total cost from shop B: - For books: \[ \text{Total Cost for Books}_B = B^T \cdot \begin{bmatrix} 225 \\ 0 \end{bmatrix} = 15 \cdot 225 + 16 \cdot 225 + 18 \cdot 225 \] \[ = 3375 + 3600 + 4050 = 11025 \] - For pens: \[ \text{Total Cost for Pens}_B = P^T \cdot \begin{bmatrix} 0 \\ 190 \end{bmatrix} = 9 \cdot 190 + 11 \cdot 190 + 15 \cdot 190 \] \[ = 1710 + 2090 + 2850 = 6650 \] - Total Cost from Shop B: \[ \text{Total Cost}_B = 11025 + 6650 = 17675 \] ### Summary of Total Costs - Total Cost from Shop A: KShs 18,375 - Total Cost from Shop B: KShs 17,675 Thus, the total costs for the three students to acquire the items from both shops are: - Shop A: KShs 18,375 - Shop B: KShs 17,675