https://file.aiquickdraw.com/mathbot/file/1726583773385v9bgj7c7.jpg

To solve the problem stated in the image, let's break it down step by step:

### 1. **Understanding the given information:**

#### Annual Sales Volume (in two markets):
- The matrix represents the sales of three products (A, B, and C) in two markets. 
- The matrix format:

\[
\text{Markets} = \begin{bmatrix} 
10000 & 2000 & 18000 \\ 
6000 & 20000 & 8000 
\end{bmatrix}
\]
  - First row: Sales in Market 1
  - Second row: Sales in Market 2
  - Columns represent products A, B, and C respectively.

#### Sale Prices per Unit:
- Product A: 8000 TK
- Product B: 9000 TK
- Product C: 7000 TK

We can represent this in a matrix form as:

\[
\text{Prices} = \begin{bmatrix} 
8000 \\ 
9000 \\ 
7000 
\end{bmatrix}
\]

#### Unit Costs per Product:
- Product A: 4000 TK
- Product B: 4500 TK
- Product C: 3000 TK

### 2. **Calculating Total Revenue in Each Market:**

To find the revenue for each product in both markets, we multiply the sales matrix with the prices matrix.

\[
\text{Revenue} = \text{Sales Matrix} \times \text{Prices Matrix}
\]

That is:

\[
\text{Revenue} = \begin{bmatrix} 
10000 & 2000 & 18000 \\ 
6000 & 20000 & 8000 
\end{bmatrix} \times \begin{bmatrix} 
8000 \\ 
9000 \\ 
7000 
\end{bmatrix}
\]

Let's compute this matrix multiplication.

### 3. **Calculating Gross Profit:**

The gross profit is calculated by subtracting the cost from the revenue. The total cost is computed by multiplying the sales matrix with the cost matrix. The cost matrix is:

\[
\text{Costs} = \begin{bmatrix} 
4000 \\ 
4500 \\ 
3000 
\end{bmatrix}
\]

Then, the **Gross Profit** can be calculated as:

\[
\text{Gross Profit} = \text{Revenue} - \text{Costs}
\]

I'll perform these calculations now.The calculations yield the following results:

1. **Total Revenue**:
   - Market 1: 224,000,000 TK
   - Market 2: 284,000,000 TK

2. **Gross Profit**:
   - Market 1: 121,000,000 TK
   - Market 2: 146,000,000 TK

These values represent the total revenue and gross profit for each market based on the given sales volumes, prices, and costs.

Would you like further details on any step or explanation? Here are some questions that might expand your understanding:

1. How are matrix operations used in business calculations like this?
2. What is the significance of separating costs from revenue when analyzing gross profit?
3. Can the matrix method be generalized to other types of products or markets?
4. How would changes in unit prices or costs affect the gross profit?
5. How can this approach help in decision-making for a manufacturer?

**Tip:** Matrix multiplication is a powerful tool for handling bulk data, making it essential in operations research and economic forecasting.

A television manufacturer produces three products A, B, and C, which are sold in two markets. The annual sales volumes are provided in a matrix, and the unit sale prices and costs of the products are given. The task is to find the total revenue in both markets and calculate the gross profit using matrix algebra.

Learn how to calculate the total revenue and gross profit for a television manufacturer selling products in two markets. This example uses matrix multiplication to analyze sales data and costs. Suitable for business students.

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