A regional health department is analyzing nutrition data from school cafeterias across three districts to evaluate and improve meal programs. They need to organize complex data about calories, nutrients, and costs across different schools and meal types. Matrix operations help them efficiently calculate totals, compare programs, and optimize budgets.

The health department collected data on three key nutrients (protein, fiber, and vitamin C) for breakfast and lunch programs across three school districts:

District A Nutrition Data (grams per meal):

• Breakfast: 12g protein, 4g fiber, 15mg vitamin C
• Lunch: 25g protein, 5g fiber, 10mg vitamin C

District B Nutrition Data (grams per meal):

• Breakfast: 15g protein, 6g fiber, 20mg vitamin C
• Lunch: 22g protein, 4g fiber, 35mg vitamin C

District C Nutrition Data (grams per meal):

• Breakfast: 10g protein, 4g fiber, 20mg vitamin C
• Lunch: 28g protein, 4g fiber, 5mg vitamin C

The department wants to find the average nutritional content when combining all three district programs and then dividing those values by [latex]3[/latex].

What if the department also tracks cost?  Each gram of protein costs $0.15, each gram of fiber costs $0.08, and each mg of vitamin C costs $0.02.

We have been organizing each district’s nutrition data as a [latex]2\times 3[/latex] matrix with rows = meals (Breakfast, Lunch) and columns = nutrients (Protein, Fiber, Vitamin C). For matrix multiplication to work, the inner dimensions must match.

• District matrix: [latex]2\times 3[/latex] (meals × nutrients.
• Price matrix: must be [latex]3\times 1[/latex] (nutrients × cost per unit.
• Product: [latex](2\times 3)(3\times 1) = 2\times 1[/latex], which returns a cost per meal (Breakfast, Lunch).

Find the cost of each meal for District A by multiplying the nutrition data with the cost matrix.

Show Solution

Nutrition data:
[latex]A= \begin{bmatrix} 12 & 4 & 15\\ 25 & 5 & 10 \end{bmatrix}[/latex]

Cost matrix:
[latex]\begin{bmatrix} 0.15\\ 0.08\\ 0.02 \end{bmatrix}[/latex]

[latex]\begin{align} A\mathbf{p} &= \begin{bmatrix} 12 & 4 & 15\\ 25 & 5 & 10 \end{bmatrix} \begin{bmatrix} 0.15\\ 0.08\\ 0.02 \end{bmatrix} \\\\ &= \begin{bmatrix} 12(0.15)+4(0.08)+15(0.02)\\[2pt] 25(0.15)+5(0.08)+10(0.02) \end{bmatrix} \\\\ &= \begin{bmatrix} 1.80+0.32+0.30\\[2pt] 3.75+0.40+0.20 \end{bmatrix}\\\\ &= \begin{bmatrix} 2.42\\[2pt] 4.35 \end{bmatrix} \end{align}[/latex]

For District A the breakfast cost is $2.42 per meal and the lunch cost is $4.35 per meal.