Hello, I need to use matrices in order to solve this problem but I am having a hard time getting started with the problem. I just need a little push to nudge me in the right direction. Please help. Thank You.

You have been hired as a consultant for Crazy Al’s Car Rentals, a new car rental agency in the city of Metropolis. Crazy Al’s Car Rentals has a total of 2500 cars that it rents from three locations within the city: Metropolis Airport, downtown, and Suburban Airport on the outskirts of the city limits.

Your research has shown the following weekly rental and return patterns:

• 17/20 of customers who rent from A return their cars to A

• 1/10 of customers who rent from A return their cars to B

• 4/5 of customers who rent from B return their cars to B

• 1/10 of customers who rent from B return their cars to A

• 1/10 of customers who rent from C return their cars to A

• 1/20 of customers who rent from C return their cars to B

Crazy Al, the owner of Crazy Al’s Car Rentals, has hired you to determine how many of his cars should be at each of his three locations at the start of each week so that the same number of cars will be there at the end of the week (and hence at the start of the next week).

To solve this problem using matrices, you can set up a system of equations based on the rental and return patterns provided.

Let's denote the number of cars at each location at the start of the week using a column vector, C = [A, B, C]. Similarly, let R = [A, B] represent the number of cars returned to each location at the end of the week.

Based on the patterns given, we can write the following equations:

For location A:
- The number of cars at A at the end of the week is the sum of those returned from A and B: R_A = (17/20)C_A + (1/10)C_B
- The number of cars at A at the start of the next week should be the same as at the start of the week: C_A = C_A

For location B:
- The number of cars at B at the end of the week is the sum of those returned from B and A: R_B = (4/5)C_B + (1/10)C_A
- The number of cars at B at the start of the next week should be the same as at the start of the week: C_B = C_B

For location C:
- The number of cars at C at the end of the week is the sum of those returned from A and B: R_C = (1/10)C_A + (1/20)C_B
- The number of cars at C at the start of the next week should be the same as at the start of the week: C_C = C_C

Now, we can rewrite these equations in matrix form as:

[R_A] [17/20 1/10 0] [C_A]
[R_B] = [1/10 4/5 0] * [C_B]
[R_C] [1/10 0 1/20] [C_C]

To find the values of C_A, C_B, and C_C, you can use matrix algebra to solve this system of equations. Let me do the calculations for you:

[R_A] [17/20 1/10 0]^-1 [C_A]
[R_B] = [1/10 4/5 0] * [C_B]
[R_C] [1/10 0 1/20] [C_C]

Here, ^-1 represents the inverse of the matrix [17/20 1/10 0].

By multiplying both sides of the equation by the inverse matrix, we can isolate [C_A, C_B, C_C] on one side:

[C_A] = [17/20 1/10 0]^-1 * [R_A]
[C_B] [1/10 4/5 0] [R_B]
[C_C] [1/10 0 1/20] [R_C]

Now, substitute the given values of [R_A, R_B, R_C] into the equation and perform the matrix multiplication to find the values of [C_A, C_B, C_C].

I hope this helps you get started with solving the problem using matrices. Let me know if you have any further questions!