For the following transaction matrix, find the gross output for each industry for the
final demands 18 and 44 units respectively.

Industry Input to Final Demand
I II
I 16 20 4
II 8 40 32

To find the gross output for each industry, we need to set up a system of linear equations using the transaction matrix and the final demands provided.

### Transaction Matrix:
\[
\begin{array}{c|c|c|c}
 & \text{Input to I} & \text{Input to II} & \text{Final Demand} \\
\hline
\text{Industry I} & 16 & 20 & 4 \\
\text{Industry II} & 8 & 40 & 32 \\
\end{array}
\]

### Let:
- \( X_1 \) be the gross output for Industry I.
- \( X_2 \) be the gross output for Industry II.

### The equations based on the transaction matrix are:

1. \( X_1 = \text{Input from Industry I to I} + \text{Input from Industry II to I} + \text{Final Demand for I} \)
   \[
   X_1 = 16X_1 + 20X_2 + 18
   \]
   
2. \( X_2 = \text{Input from Industry I to II} + \text{Input from Industry II to II} + \text{Final Demand for II} \)
   \[
   X_2 = 8X_1 + 40X_2 + 44
   \]

Now, we can express these equations as:

\[
\begin{aligned}
X_1 & = 16X_1 + 20X_2 + 18 \\
X_2 & = 8X_1 + 40X_2 + 44 \\
\end{aligned}
\]

Rearrange the equations to solve for \( X_1 \) and \( X_2 \):

\[
\begin{aligned}
X_1 - 16X_1 - 20X_2 & = 18 \\
X_2 - 8X_1 - 40X_2 & = 44 \\
\end{aligned}
\]

Simplify these to:

\[
\begin{aligned}
-15X_1 - 20X_2 & = 18 \quad \text{(Equation 1)} \\
-8X_1 - 39X_2 & = 44 \quad \text{(Equation 2)} \\
\end{aligned}
\]

Now, we can solve this system of linear equations using matrix algebra or substitution/elimination methods.

Let me calculate the values of \( X_1 \) and \( X_2 \).The solution to the system of equations gives us the following:

- \( X_1 = \frac{178}{425} \) (approximately 0.419)
- \( X_2 = \frac{-516}{425} \) (approximately -1.214)

However, the negative value for \( X_2 \) suggests there might be an inconsistency or issue with the problem setup, as gross output should typically be non-negative.

Would you like to delve deeper into this result or explore any related concepts?

### Follow-up Questions:
1. Would you like to verify the calculations for any potential errors?
2. Should we review the setup of the problem, especially the equations derived?
3. Are you interested in learning how this transaction matrix method is applied in economics?
4. Would you like to understand the significance of negative output in this context?
5. Should we explore different methods to solve this system?

**Tip:** When solving systems of linear equations, especially in applied contexts, it's important to check the assumptions and consistency of the setup to ensure the results are meaningful.

Explore how to solve systems of linear equations using transaction matrices. Learn step-by-step methods to find gross outputs for industries based on given final demands and transaction matrix inputs. Includes detailed calculations and implications of negative outputs.

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