The input-output matrix \(\mathbf{C}\) is derived from the inter-industry transaction table provided in the problem. This matrix represents the input coefficients, indicating how much of each sector's output is required to produce one unit of output in each sector.

### Steps to Derive \(\mathbf{C}\):

1. **Understand the Transaction Table:**
   The transaction table you provided describes the flow of goods and services between different sectors of the economy. Each entry \(a_{ij}\) in the table represents the amount of output from sector \(i\) that is used as an input by sector \(j\).

   Here's the table again for reference:

   \[
   \begin{array}{ c|c c c |c }
   & \text{Agriculture} & \text{Manufactures} & \text{Services} & \text{Total Output} \\
   \hline
   \text{Agriculture} & 60 & 180 & 0 & 240 \\
   \text{Manufactures} & 30 & 60 & 30 & 120 \\
   \text{Services} & 0 & 40 & 10 & 50 \\
   \text{Final Demand} & 150 & 160 & 180 & \\
   \end{array}
   \]

   - The rows correspond to the sectors supplying inputs.
   - The columns correspond to the sectors using those inputs.

2. **Calculate the Total Output for Each Sector:**
   The total output for each sector is obtained by summing up the elements in each column:

   - Agriculture: \( 60 + 30 + 0 + 150 = 240 \)
   - Manufactures: \( 180 + 60 + 40 + 160 = 440 \)
   - Services: \( 0 + 30 + 10 + 180 = 220 \)

3. **Compute Input Coefficients:**
   The elements of the matrix \(\mathbf{C}\) are calculated as the ratio of each inter-industry transaction to the total output of the sector that receives the input. This gives us the input coefficients, which represent the proportion of input required from one sector to produce a unit of output in another sector.

   For example, to calculate the element \(c_{11}\) (input from Agriculture to Agriculture):
   
   \[
   c_{11} = \frac{\text{Input from Agriculture to Agriculture}}{\text{Total Output of Agriculture}} = \frac{60}{240} = 0.25
   \]

   Similarly, calculate other coefficients:

   \[
   \mathbf{C} = \begin{pmatrix}
   \frac{60}{240} & \frac{180}{440} & \frac{0}{220} \\
   \frac{30}{240} & \frac{60}{440} & \frac{30}{220} \\
   \frac{0}{240} & \frac{40}{440} & \frac{10}{220}
   \end{pmatrix}
   \approx \begin{pmatrix}
   0.25 & 0.41 & 0.00 \\
   0.125 & 0.136 & 0.136 \\
   0.00 & 0.091 & 0.045
   \end{pmatrix}
   \]

This matrix \(\mathbf{C}\) shows the direct input requirements of each sector. Each element \(c_{ij}\) reflects how much output from sector \(i\) is needed to produce one unit of output in sector \(j\). 

### Summary of the Steps:

1. **Obtain the inter-industry transaction values** from the table.
2. **Determine the total output** for each sector.
3. **Calculate each element of \(\mathbf{C}\)** as the ratio of inter-industry transaction to the total output of the receiving sector.

Would you like further details or clarifications? Here are five related questions:

1. What does each element of the \(\mathbf{C}\) matrix represent economically?
2. How does changing the total output values affect the matrix \(\mathbf{C}\)?
3. Can the matrix \(\mathbf{C}\) be used to analyze the impacts of economic shocks?
4. How would the introduction of a new sector affect the matrix \(\mathbf{C}\)?
5. What assumptions underlie the use of an input-output matrix like \(\mathbf{C}\) in economic analysis?

### Tip:
Always double-check your matrix calculations, especially when dealing with ratios, as small errors can significantly impact the analysis.

Explore how the input-output matrix \(\mathbf{C}\) is derived and used in economic analysis. Learn about the economic significance of each element in the matrix, its calculation method, and applications in analyzing economic interdependencies. Ideal for students and researchers interested in economic modeling and analysis.

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