Hello, I need to use a matrix to solve this but am stumped on how to do it. All I need is a little nudge to push 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: A, B, c.

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

• 85% of customers who rent from A return their cars to A

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

• 80% of customers who rent from B return their cars to B

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

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

• 5% 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).

Use what you have learned about systems of equations and matrices to set up a system of equations representing this situation and solve this problem. Then prepare a brief report outlining your work, your methods, and your findings for Crazy Al.

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!

To solve this problem using matrices, we can represent the number of cars at each location using a matrix. Let's call this matrix X.

X = [a, b, c]

Where a, b, and c represent the number of cars at locations A, B, and C, respectively.

Now, let's consider the rental and return patterns given:

1. 85% of customers who rent from A return their cars to A:
This means that 85% of the cars rented from A will stay at A. So, 85% of the cars rented from A will add to the existing number of cars at A.

So, the number of cars at A, denoted by a, can be represented as:
a = a + 0.85*a

2. 10% of customers who rent from A return their cars to B:
This means that 10% of the cars rented from A will be returned to B. So, 10% of cars rented from A will be subtracted from A and added to B.

So, the number of cars at B, denoted by b, can be represented as:
b = b + 0.10*a

3. 80% of customers who rent from B return their cars to B:
This means that 80% of the cars rented from B will stay at B. So, 80% of the cars rented from B will add to the existing number of cars at B.

So, the number of cars at B, denoted by b, can be represented as:
b = b + 0.80*b

4. 10% of customers who rent from B return their cars to A:
This means that 10% of the cars rented from B will be returned to A. So, 10% of cars rented from B will be subtracted from B and added to A.

So, the number of cars at A, denoted by a, can be represented as:
a = a + 0.10*b

5. 10% of customers who rent from C return their cars to A:
This means that 10% of the cars rented from C will be returned to A. So, 10% of cars rented from C will be subtracted from C and added to A.

So, the number of cars at A, denoted by a, can be represented as:
a = a + 0.10*c

6. 5% of customers who rent from C return their cars to B:
This means that 5% of the cars rented from C will be returned to B. So, 5% of cars rented from C will be subtracted from C and added to B.

So, the number of cars at B, denoted by b, can be represented as:
b = b + 0.05*c

Now we have a system of equations based on the rental and return patterns. We can represent this system of equations using a matrix equation:

X = AX

Where A is another matrix representing the coefficients of the variables in the equations. The elements of matrix A can be determined from the coefficients in the above equations.

Solving this system of equations will give us the values of a, b, and c, which represent the number of cars at each location.

I hope this gives you a nudge in the right direction to set up the matrix equations for this problem. Let me know if you need further assistance!