Question 1197518
**Part i**

**Transition Matrix**

* **Define States:**
    * State 1: Purchased Brand A
    * State 2: Purchased Brand B
    * State 3: Purchased Neither Brand

* **Transition Probabilities:**

| State | State 1 (A) | State 2 (B) | State 3 (Neither) |
|---|---|---|---|
| **State 1 (A)** | 0.80 | 0.15 | 0.05 |
| **State 2 (B)** | 0.12 | 0.85 | 0.03 |
| **State 3 (Neither)** | 0.20 | 0.15 | 0.65 |

**State Matrix (Initial Distribution)**

* **State 1 (A):** 20,000 / 100,000 = 0.2 
* **State 2 (B):** 35,000 / 100,000 = 0.35
* **State 3 (Neither):** 45,000 / 100,000 = 0.45

* **Initial State Matrix:** 
   [0.2 0.35 0.45]

**Part ii**

**Determine the number of customers in each brand after 2 weeks**

* **Multiply the Initial State Matrix by the Transition Matrix twice:**

   * **After 1 week:** 
      [0.2 0.35 0.45] * 
      [ 0.80 0.15 0.05 
        0.12 0.85 0.03 
        0.20 0.15 0.65 ] 

   * **After 2 weeks:** 
      [Result from 1st week] * 
      [ 0.80 0.15 0.05 
        0.12 0.85 0.03 
        0.20 0.15 0.65 ] 

* **Calculate the resulting state matrix.** 
* **Multiply each state probability by the total population (100,000) to find the number of customers in each brand after 2 weeks.**

**Part b**

**Determine the machinery, electricity, and oil available for export**

Let:

* X: Machinery output
* Y: Electricity output
* Z: Oil output

We have the following system of equations:

* X = 3000 - 0.1Y - 0.2Z 
* Y = 5000 - 0.3X - 0.2Z
* Z = 2000 - 0.3X - 0.1Y

**Solve this system of equations to find the values of X, Y, and Z.**

* **Method 1: Substitution or Elimination**
    * Solve for one variable in one equation and substitute it into the other equations. 
    * Repeat until you find the values of all three variables.

* **Method 2: Matrix Method**
    * Represent the system of equations in matrix form and use matrix inversion to solve for X, Y, and Z.

**Once you find the values of X, Y, and Z:**

* **Machinery available for export:** Total machinery output (3000) - Machinery used as input (value of X)
* **Electricity available for export:** Total electricity output (5000) - Electricity used as input (value of Y)
* **Oil available for export:** Total oil output (2000) - Oil used as input (value of Z)

This will give you the amounts of machinery, electricity, and oil available for export after accounting for the internal consumption within the country.

**Note:**

* This analysis assumes a closed economic system where all production and consumption occur within the country.
* In reality, there may be imports and exports of these goods, which would affect the final available quantities.