Matrix *A* shows the number of servings of each food that each person eats per day. Matrix *B* shows the number of units of each nutrient in a serving of each food.
 
*A* =
 
 
Chris

Fred

Kate
 
 
pork 

 beans 

 pasta

1.5 

 0 

 0.5

0 

 0 

 1

1.5 

 1 

 2
 
 
*B* =
 
 
pork

beans

pasta
 
 
Zn 

 E 

 B12

8 

 16 

 16

0 

 18 

 2

12 

 0 

 14
 
Find the product matrix *AB* and label the rows and columns.


 
 
 
 
What does the product matrix AB represent?

The total daily amount of each nutrient for each person.The total amount of each food each person has daily.    The product is meaningless.The total amount of each nutrient for each food.The total daily amount of each food each person eats.

We are given two matrices: matrix *A* (which represents the number of servings of food consumed per day by each person) and matrix *B* (which represents the number of units of nutrients in a serving of each food). To find the product matrix *AB*, we need to multiply matrix *A* by matrix *B*.

Matrix *A*:
\[
A =
\begin{pmatrix}
1.5 & 0 & 0.5 \\
0 & 0 & 1 \\
1.5 & 1 & 2
\end{pmatrix}
\]
Where rows correspond to Chris, Fred, and Kate, and columns correspond to pork, beans, and pasta.

Matrix *B*:
\[
B =
\begin{pmatrix}
8 & 16 & 16 \\
0 & 18 & 2 \\
12 & 0 & 14
\end{pmatrix}
\]
Where rows correspond to pork, beans, and pasta, and columns correspond to nutrients Zn (zinc), E (vitamin E), and B12.

Now, we will calculate the product matrix *AB* by multiplying the rows of *A* with the columns of *B*:

\[
AB = A \times B =
\begin{pmatrix}
1.5 & 0 & 0.5 \\
0 & 0 & 1 \\
1.5 & 1 & 2
\end{pmatrix}
\times
\begin{pmatrix}
8 & 16 & 16 \\
0 & 18 & 2 \\
12 & 0 & 14
\end{pmatrix}
\]

### Row 1 (Chris):
\[
(1.5 \times 8) + (0 \times 0) + (0.5 \times 12) = 12 + 0 + 6 = 18 \quad \text{(Zn)}
\]
\[
(1.5 \times 16) + (0 \times 18) + (0.5 \times 0) = 24 + 0 + 0 = 24 \quad \text{(E)}
\]
\[
(1.5 \times 16) + (0 \times 2) + (0.5 \times 14) = 24 + 0 + 7 = 31 \quad \text{(B12)}
\]

### Row 2 (Fred):
\[
(0 \times 8) + (0 \times 0) + (1 \times 12) = 0 + 0 + 12 = 12 \quad \text{(Zn)}
\]
\[
(0 \times 16) + (0 \times 18) + (1 \times 0) = 0 + 0 + 0 = 0 \quad \text{(E)}
\]
\[
(0 \times 16) + (0 \times 2) + (1 \times 14) = 0 + 0 + 14 = 14 \quad \text{(B12)}
\]

### Row 3 (Kate):
\[
(1.5 \times 8) + (1 \times 0) + (2 \times 12) = 12 + 0 + 24 = 36 \quad \text{(Zn)}
\]
\[
(1.5 \times 16) + (1 \times 18) + (2 \times 0) = 24 + 18 + 0 = 42 \quad \text{(E)}
\]
\[
(1.5 \times 16) + (1 \times 2) + (2 \times 14) = 24 + 2 + 28 = 54 \quad \text{(B12)}
\]

Thus, the product matrix *AB* is:
\[
AB =
\begin{pmatrix}
18 & 24 & 31 \\
12 & 0 & 14 \\
36 & 42 & 54
\end{pmatrix}
\]

### Labeling the rows and columns:

The rows represent the people (Chris, Fred, Kate), and the columns represent the nutrients (Zn, E, B12). Therefore, the product matrix *AB* looks like this:

\[
\begin{array}{c|ccc}
& \text{Zn} & \text{E} & \text{B12} \\
\hline
\text{Chris} & 18 & 24 & 31 \\
\text{Fred} & 12 & 0 & 14 \\
\text{Kate} & 36 & 42 & 54 \\
\end{array}
\]

### Interpretation:
The product matrix *AB* represents **the total daily amount of each nutrient (Zn, E, B12) for each person (Chris, Fred, Kate)** based on their consumption of pork, beans, and pasta.

### Correct answer:
**The total daily amount of each nutrient for each person.**

---

Would you like more details or have any questions?

Here are five related questions:
1. How do you compute matrix products in general?
2. What properties of matrices impact the order of multiplication?
3. How does a zero entry in matrix *A* or *B* affect the product?
4. Can matrix multiplication be used for other types of nutrient tracking scenarios?
5. What are the possible real-world applications of matrix multiplication?

**Tip:** When multiplying matrices, make sure the number of columns in the first matrix equals the number of rows in the second.

Matrix *A* shows the number of servings of each food that each person eats per day. Matrix *B* shows the number of units of each nutrient in a serving of each food.

*A* =

Chris | Fred | Kate

pork | beans | pasta

1.5 | 0 | 0.5
0 | 0 | 1
1.5 | 1 | 2

*B* =

pork | beans | pasta

Zn | E | B12

8 | 16 | 16
0 | 18 | 2
12 | 0 | 14

Find the product matrix *AB* and label the rows and columns.

What does the product matrix AB represent?

Learn how to compute the product of matrices to determine the total daily amount of each nutrient (Zn, E, B12) consumed by Chris, Fred, and Kate based on their food intake. This example covers matrix multiplication, nutrient tracking, and linear algebra. Suitable for Grades 10-12.

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