A dealership has new and used cars and trucks. The following matrices give the number of vehicles sold in​ January, February, and March. Write a matrix that expresses the average vehicle sales for the first quarter of the year.

January

February

March

nbsp Cars Cars

nbsp Trucks Trucks

nbsp Cars Cars

nbsp Trucks Trucks

nbsp Cars Cars

nbsp Trucks Trucks

New

 left bracket Start 2 By 2 Matrix 1st Row 1st Column 15 2nd Column 12 2nd Row 1st Column 22 2nd Column 17 EndMatrix right bracket

15

12

22

17

 left bracket Start 2 By 2 Matrix 1st Row 1st Column 15 2nd Column 13 2nd Row 1st Column 32 2nd Column 23 EndMatrix right bracket

15

13

32

23

 left bracket Start 2 By 2 Matrix 1st Row 1st Column 24 2nd Column 14 2nd Row 1st Column 36 2nd Column 38 EndMatrix right bracket

24

14

36

38

Used

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Part 1

A matrix that expresses the average vehicle sales for the first quarter of the year is 

enter your response here.

Let's break down the problem to create a matrix that expresses the average vehicle sales for the first quarter of the year.

We are provided with matrices for the number of **New** and **Used** cars and trucks sold in January, February, and March. These matrices are:

**January (New Vehicles)**  
\[
\begin{bmatrix}
15 & 12 \\
22 & 17
\end{bmatrix}
\]

**February (New Vehicles)**  
\[
\begin{bmatrix}
15 & 13 \\
32 & 23
\end{bmatrix}
\]

**March (New Vehicles)**  
\[
\begin{bmatrix}
24 & 14 \\
36 & 38
\end{bmatrix}
\]

### Step 1: Add the Matrices Together

We will first add the corresponding elements from the matrices for January, February, and March. For each position, we add the values:

**New Cars:**  
For the first row and first column (new cars), we have:  
\( 15 + 15 + 24 = 54 \)

For the first row and second column (new trucks), we have:  
\( 12 + 13 + 14 = 39 \)

**Used Cars:**  
For the second row and first column (used cars), we have:  
\( 22 + 32 + 36 = 90 \)

For the second row and second column (used trucks), we have:  
\( 17 + 23 + 38 = 78 \)

So, the sum matrix for the first quarter is:
\[
\begin{bmatrix}
54 & 39 \\
90 & 78
\end{bmatrix}
\]

### Step 2: Find the Average for the Quarter

To find the average vehicle sales for the first quarter, divide each element by 3 (since there are 3 months in the quarter):

\[
\frac{1}{3} \times 
\begin{bmatrix}
54 & 39 \\
90 & 78
\end{bmatrix}
=
\begin{bmatrix}
18 & 13 \\
30 & 26
\end{bmatrix}
\]

### Final Answer

The matrix that expresses the **average vehicle sales for the first quarter** is:
\[
\boxed{
\begin{bmatrix}
18 & 13 \\
30 & 26
\end{bmatrix}
}
\]

This matrix shows the average number of new cars, new trucks, used cars, and used trucks sold per month during the first quarter.

Let me know if you have any questions or would like more details!

### 5 Related Questions:
1. How do you find the sum of two matrices?
2. What are the steps to calculate the average of elements in a matrix?
3. How would the result change if the data were for 4 months instead of 3?
4. Can we extend this method to find average sales for different quarters?
5. How can we interpret matrix results in terms of real-world sales data?

### Tip: Always ensure that matrices are of the same dimensions before adding or averaging them.

A dealership has new and used cars and trucks. The following matrices give the number of vehicles sold in January, February, and March. Write a matrix that expresses the average vehicle sales for the first quarter of the year.

This problem explores how to calculate the average vehicle sales for the first quarter of the year using matrix addition and division. We add the number of new and used cars and trucks sold in January, February, and March and divide by 3 to find the average. Suitable for students in Grades 9-12 learning about matrices and averages.

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