Skill: Interpret a matrix product in context

A matrix product combines matching categories. To interpret it, check that the columns of the first matrix and the rows of the second matrix refer to the same categories in the same order. Each result is found by multiplying corresponding entries and adding the products.

Suppose a bakery uses three ingredients to make two products. The matrix $Q$ gives the kilograms of each ingredient needed for one batch of each product:

$$Q= \begin{bmatrix} 2 & 1 & 0.5\\ 1 & 2 & 1.5 \end{bmatrix} \quad \begin{array}{l} \text{Product A}\\ \text{Product B} \end{array}$$

The columns are flour, sugar, and cocoa, in that order. Their costs per kilogram are:

$$c= \begin{bmatrix} 1.20\\ 0.80\\ 2.00 \end{bmatrix} \quad\text{dollars per kilogram}$$

To find the ingredient cost per batch for each product, multiply $Q$ by $c$. The dimensions are $2\times3$ and $3\times1$, so the product has dimensions $2\times1$—one cost for each product.

$$Qc= \begin{bmatrix} 2(1.20)+1(0.80)+0.5(2.00)\\ 1(1.20)+2(0.80)+1.5(2.00) \end{bmatrix} = \begin{bmatrix} 4.20\\ 5.80 \end{bmatrix}$$

For Product A, the first entry adds the flour, sugar, and cocoa costs: 4.20$. The second entry gives a cost of$$5.80$ for Product B. Since kilograms per batch multiplied by dollars per kilogram gives dollars per batch, the entries represent ingredient cost per batch for each product.