The double matrix method can be used when you have a question that asks you to segment a population twice in two different ways. It can be a great alternative to Venn diagrams.

The following example walks you through the approach to a typical “double matrix” question.

*A parking lot contains 80 vehicles. Each vehicle is either a car or a truck, and each vehicle is either red or green. 35 vehicles are red, and 60 vehicles are cars. If there are 9 green trucks, how many red cars are there?*

(A) 11

(B) 24

(C) 28

(D) 36

(E) 45

The first step is recognizing that this question is a “double matrix” question. You have a population of vehicles that can be segmented twice in two ways each: (1) into cars and trucks and (2) into red and green. Note the following statement from the question:

*Each vehicle is either a car or a truck, and each vehicle is either red or green.*

Double matrix questions all take the following format:

Based on the segments in the question you can setup the following matrix:

Using the information in the question, fill in as much of the template as you can. For this question, you know the following:

- A parking lot contains 80 vehicles
- 35 vehicles are red
- 60 vehicles are cars
- There are 9 green trucks

Therefore you can create the following matrix:

Now you can begin filling in some additional information, such as total green vehicles and total trucks, using simple algebra.

*35 red vehicles + # of green vehicles = 80
Subtract 35 from each side: # of green vehicles = 45
*

**There are 45 green vehicles**

*Total trucks: 60 cars + # of trucks = 80
Subtract 60 from each side: # of trucks = 20
*

**There are 20 trucks**

You can now add the new findings into the matrix:

Again, with some more basic algebra you can complete more of the matrix.

*# of red trucks + 9 green trucks = 20 trucks
Subtract 9 from each side: # of red trucks = 11
*

**There are 11 red trucks**

With one more step, you can calculate the number of red cars:

*# of red cars + 11 red trucks = 35 red vehicles
Subtract 11 from each side: # of red trucks = 24
*

**There are 24 red cars**

Note that you can verify your answer by calculating the # of green cars, completing the matrix.

**The final answer is B, 24.**

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