To solve this problem using matrices, we can set up a system of equations based on the given weekly rental and return patterns.
Let's define the following variables:
- A: Number of cars at location A at the start of the week
- B: Number of cars at location B at the start of the week
- C: Number of cars at location C at the start of the week
Now, let's analyze the rental patterns:
1. 85% of customers who rent from A return their cars to A:
This means that 85% of cars rented from A would stay at A. So, 0.85A cars would be returned to A.
2. 10% of customers who rent from A return their cars to B:
This means that 10% of cars rented from A would be returned to B. So, 0.1A cars would be returned to B.
3. 80% of customers who rent from B return their cars to B:
This means that 80% of cars rented from B would stay at B. So, 0.8B cars would be returned to B.
4. 10% of customers who rent from B return their cars to A:
This means that 10% of cars rented from B would be returned to A. So, 0.1B cars would be returned to A.
5. 10% of customers who rent from C return their cars to A:
This means that 10% of cars rented from C would be returned to A. So, 0.1C cars would be returned to A.
6. 5% of customers who rent from C return their cars to B:
This means that 5% of cars rented from C would be returned to B. So, 0.05C cars would be returned to B.
Now we can set up the system of equations:
Equation 1: 0.85A + 0.1B + 0.1C = A
Equation 2: 0.1A + 0.8B + 0.05C = B
Equation 3: 0.1C + 0.05C = C
These equations represent the total number of cars at each location after the return patterns.
To solve this system of equations, we can write it in matrix form:
| 0.85 0.1 0.1 | | A | | A |
| 0.1 0.8 0.05 | x | B | = | B |
| 0 0 1 | | C | | C |
Where the left matrix represents the coefficients of the variables, and the right matrix represents the variables themselves.
To solve the system, you can use matrix operations to find the values of A, B, and C.
I hope this helps you get started on solving the problem. If you have any further questions, feel free to ask!