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To address the problem based on the provided matrices, we need to calculate:

1. **Total expected demand** for each final product.
2. **Quantities needed of each of the four components**.

Here's the step-by-step breakdown:

### Part (a): Total expected demand for each final product
We are given the matrix \( S \), which represents sales across three regions for five products. To find the total expected demand for each final product, sum the entries for each product across the regions.

#### Matrix \( S \) (Sales across Regions):

\[
S = \begin{bmatrix}
500 & 200 & 350 \\
400 & 300 & 100 \\
250 & 425 & 50 \\
100 & 150 & 350 \\
200 & 175 & 225
\end{bmatrix}
\]

- Total demand for product 1: \( 500 + 200 + 350 = 1050 \)
- Total demand for product 2: \( 400 + 300 + 100 = 800 \)
- Total demand for product 3: \( 250 + 425 + 50 = 725 \)
- Total demand for product 4: \( 100 + 150 + 350 = 600 \)
- Total demand for product 5: \( 200 + 175 + 225 = 600 \)

Thus, the total expected demand vector for the products is:

\[
\text{Total Demand} = \begin{bmatrix} 1050 \\ 800 \\ 725 \\ 600 \\ 600 \end{bmatrix}
\]

### Part (b): Quantities needed of each of the four components
The matrix \( R \) represents the number of units of each component used in producing each product. To determine the total quantities needed for each component, we multiply the total demand for each product by the corresponding entries in matrix \( R \).

#### Matrix \( R \) (Components used for each product):

\[
R = \begin{bmatrix}
1 & 0 & 2 & 0 \\
1 & 1 & 1 & 0 \\
2 & 1 & 0 & 3 \\
0 & 2 & 1 & 4 \\
1 & 2 & 3 & 1
\end{bmatrix}
\]

We will now compute the total quantities needed for each component by summing the contributions from each product's total demand.

Let me calculate these values for you now.The total quantities needed of each of the four components are as follows:

- Component 1: 3900 units
- Component 2: 3925 units
- Component 3: 5300 units
- Component 4: 5175 units

Let me know if you would like further clarifications or additional details!

Here are some related questions:
1. How would changes in regional demand affect the total quantities of components?
2. How would the cost of each component affect the overall production cost?
3. What if an additional product is introduced? How would we adjust these matrices?
4. How does the labor hour cost in Matrix P affect total production expenses?
5. Can we calculate the profit if given the selling price of each product?

**Tip**: When working with matrices, always ensure the dimensions are compatible before performing operations like multiplication.

A company manufactures five products. Given the sales matrix S, component usage matrix R, and resource matrix P, calculate (a) the total expected demand for each final product and (b) the quantities needed of each of the four components.

This problem involves calculating the total demand for five products across three sales regions using matrix operations. Additionally, it calculates the required quantities of four components using the component usage matrix and total demand.

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