Math Problem Statement
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. Find the product matrix AB, label the rows and columns, and explain what the product matrix AB represents.
Solution
To solve the problem in the uploaded image, we will compute the matrix product . Let me provide the solution step by step:
Matrix (Servings per Day)
\begin{bmatrix} 0.5 & 1 & 0.5 \\ 0.5 & 0 & 0.5 \\ 1 & 1 & 0 \end{bmatrix}$$ Rows correspond to Betty, Fred, and Ian. Columns correspond to beef, carrots, and bread. ### Matrix $$B$$ (Nutrients per Serving) $$B = \begin{bmatrix} 8 & 18 & 2 \\ 20 & 2 & 16 \\ 2 & 4 & 6 \end{bmatrix}$$ Rows correspond to beef, carrots, and bread. Columns correspond to nutrients (Cu, D, B12). ### Compute the Product $$AB$$ The product $$AB$$ will have dimensions $$3 \times 3$$, where each entry is computed as: $$(AB)_{ij} = \sum_{k} A_{ik} \cdot B_{kj}$$ The resulting matrix will represent the total daily amount of each nutrient (Cu, D, B12) consumed by each person. #### Calculation of Each Entry For each person (row in $$A$$) and nutrient (column in $$B$$): - **Betty's Nutrients**: - Cu: $$0.5 \times 8 + 1 \times 20 + 0.5 \times 2 = 4 + 20 + 1 = 25$$ - D: $$0.5 \times 18 + 1 \times 2 + 0.5 \times 4 = 9 + 2 + 2 = 13$$ - B12: $$0.5 \times 2 + 1 \times 16 + 0.5 \times 6 = 1 + 16 + 3 = 20$$ - **Fred's Nutrients**: - Cu: $$0.5 \times 8 + 0 \times 20 + 0.5 \times 2 = 4 + 0 + 1 = 5$$ - D: $$0.5 \times 18 + 0 \times 2 + 0.5 \times 4 = 9 + 0 + 2 = 11$$ - B12: $$0.5 \times 2 + 0 \times 16 + 0.5 \times 6 = 1 + 0 + 3 = 4$$ - **Ian's Nutrients**: - Cu: $$1 \times 8 + 1 \times 20 + 0 \times 2 = 8 + 20 + 0 = 28$$ - D: $$1 \times 18 + 1 \times 2 + 0 \times 4 = 18 + 2 + 0 = 20$$ - B12: $$1 \times 2 + 1 \times 16 + 0 \times 6 = 2 + 16 + 0 = 18$$ ### Resultant Matrix $$AB$$ $$AB = \begin{bmatrix} 25 & 13 & 20 \\ 5 & 11 & 4 \\ 28 & 20 & 18 \end{bmatrix}$$ - Rows: Betty, Fred, Ian. - Columns: Cu, D, B12. ### Interpretation of $$AB$$ The product matrix $$AB$$ represents **the total daily amount of each nutrient for each person**. --- Would you like further clarifications, or do you have additional questions? Here are 5 possible follow-ups: 1. How does matrix multiplication work in general? 2. Can you verify the result step-by-step for one specific row? 3. How could this type of analysis be extended to include more foods or nutrients? 4. What other real-life scenarios can matrix multiplication be applied to? 5. How do row and column labels influence the interpretation of matrix operations? **Tip:** Always check the dimensions of matrices before multiplying them to ensure compatibility!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Linear Algebra
Matrix Representation
Formulas
Matrix Multiplication Formula: (AB)_{ij} = ∑_{k} A_{ik} * B_{kj}
Theorems
Matrix Multiplication Rule
Suitable Grade Level
Grades 10-12
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