Chapter 4.4, Problem 71E

Cereal. A nutritionist for a cereal company blends two cereals in three different mixes. The amounts of protein, carbohydrate and fat (in grams per ounce) in each cereal are given by matrix $M$. The amounts of each cereal used in the three mixes are given by matrix $N$.

$\begin{array}{l}\begin{array}{ccc}& \text{ Cereal}\text{A}& \text{Cereal}\text{B}\end{array}\\ M=\left[\begin{array}{cc}\text{4}\text{g}/\text{oz}& \text{2}\text{g}/\text{oz}\\ \begin{array}{l}\text{20}\text{g}/\text{oz}\\ \text{3}\text{g}/\text{oz}\end{array}& \begin{array}{l}\text{16}\text{g}/\text{oz}\\ \text{1}\text{g}/\text{oz}\end{array}\end{array}\right]\begin{array}{l}\text{Protein}\hfill \\ \text{Carbohydrate}\hfill \\ \text{Fat}\hfill \end{array}\end{array}$

$\begin{array}{l}\begin{array}{cccc}& \text{ Mix}\text{X}& \text{Mix}\text{Y}& \text{Mix}\text{Z}\end{array}\\ N=\left[\begin{array}{ccc}\begin{array}{l}\text{15oz}\\ \text{5oz}\end{array}& \begin{array}{l}\text{10oz}\\ \text{10oz}\end{array}& \begin{array}{l}\text{ 5oz}\\ \text{ 15oz}\end{array}\end{array}\right]\begin{array}{c}\text{Cereal}\text{A}\\ \text{Cereal}\text{B}\end{array}\end{array}$

(A) Find the amount of protein in mix $X$.

(B) Find the amount of fat in mix $Z$ .

(C) Discuss possible interpretations of the elements in the matrix products $MN$ and $NM$.

(D) If either of the products $MN$ or $NM$ has a meaningful interpretation. Find the product and label its rows and columns.