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### Problem Breakdown:

The question describes the sales of four products (rice, sugar, eggs, noodles) by three stores (A, B, and C) in one week, followed by three sub-questions (a, b, and c) involving matrix operations.

Given data### Problem Breakdown:

The question describes the sales of four products (rice, sugar, eggs, noodles) by three stores (A, B, and C) in one week, followed by three sub-questions (a, b, and c) involving matrix operations.

Given 

- **Store A**: 
  - 40 kg of rice
  - 50 kg of sugar
  - 40 kg of eggs
  - 60 kg of noodles
- **Store B**:
  - 50 kg of rice
  - 60 kg of sugar
  - 50 kg of eggs
  - 40 kg of noodles
- **Store C**:
  - 80 kg of rice
  - 60 kg of sugar
  - 40 kg of eggs
  - 30 kg of noodles

### Let's break it down step by step:

---

### (a) **Matrix of Sales for Each Store by Product**

We are asked to form a \( 3 \times 4 \) matrix to summarize the sales for each store.

- Store A: \([40, 50, 40, 60]\)
- Store B: \([50, 60, 50, 40]\)
- Store C: \([80, 60, 40, 30]\)

So, the matrix \( M \) is:

\[
M = \begin{pmatrix}
40 & 50 & 40 & 60 \\
50 & 60 & 50 & 40 \\
80 & 60 & 40 & 30
\end{pmatrix}
\]

---

### (b) **Matrix After 10% Increase in Sales**

The question asks us to calculate the sales for the next week, where sales increase by 10%. This can be done by multiplying each element of matrix \( M \) by 1.10 (to account for the 10% increase).

Let me calculate that.After a 10% increase, the sales matrix for the next week is:

\[
M_{\text{next week}} = \begin{pmatrix}
44 & 55 & 44 & 66 \\
55 & 66 & 55 & 44 \\
88 & 66 & 44 & 33
\end{pmatrix}
\]

---

### (c) **Matrix of Total Income for Two Weeks**

The prices for each product are given as:
- Rice: S/. 4 per kg
- Sugar: S/. 3 per kg
- Eggs: S/. 5 per kg
- Noodles: S/. 3 per kg

We now need to find the total income for each store for both weeks combined. This can be done by first summing the original and increased sales, then multiplying each product by its price. The prices are given as a vector: \([4, 3, 5, 3]\).

Let's compute this income matrix.The total income for the two weeks in each store is:

- **Store A**: S/. 1449
- **Store B**: S/. 1575
- **Store C**: S/. 1659

---

Would you like more details or clarifications on any step?

Here are some additional related questions:
1. How is matrix multiplication applied to calculate income in different scenarios?
2. What would happen if the prices of products change?
3. How would you handle a decrease in sales by 10% instead of an increase?
4. Can you explore the average sales per product across all stores?
5. How would the calculations change if there were a different set of products?

**Tip**: When dealing with matrices, ensure that the dimensions are compatible for operations like multiplication or addition to avoid errors.

En un mercado del distrito de Chilca hay tres tiendas de abarrotes A, B y C que venden cuatro productos: arroz, azúcar, huevos y fideos. En una semana, la tienda A vendió 40 kg de arroz, 50 kg de azúcar, 40 kg de huevos y 60 kg de fideos; la tienda B vendió 50 kg de arroz, 60 kg de azúcar, 50 kg de huevos y 40 kg de fideos; y la tienda C vendió 80 kg de arroz, 60 kg de azúcar, 40 kg de huevos y 30 kg de fideos.

(a) Use una matriz de orden 3x4 para resumir la información sobre las ventas de las tres tiendas, por tienda y por producto.
(b) Si en la semana siguiente las ventas en las tres tiendas aumentan un 10%, halle la matriz de ventas para esa semana.
(c) Si el precio de los productos en las tres tiendas es: S/. 4 el kg de arroz, S/. 3 el kg de azúcar, S/. 5 el kg de huevos y S/. 3 el kg de fideos, halle una matriz del orden 3x1 que represente los ingresos totales en el periodo de las dos semanas en cada una de las tiendas.

Solve a matrix problem involving sales of four products across three stores. Learn how to summarize sales data, apply a 10% sales increase, and calculate total revenue using matrix multiplication. Includes step-by-step explanation and matrix calculations for algebra students in Grades 9-12.

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