Chapter 4, Problem 44RE

Material costs. A manufacturer wishes to make two different bronze alloys in a metal foundry. The quantities of copper, tin and zinc needed are indicated in matrix $M$.The costs for these materials (in dollars per pound) from two suppliers are summarized in matrix $N$. The company must choose one supplier or the other.

$\begin{array}{l}\begin{array}{ccc}\text{Cooper}& \text{Tin}& \text{Zinc}\end{array}\\ M=\left[\begin{array}{ccc}4.800\text{lb}& 600\text{lb}& 300\text{lb}\\ 6,000\text{lb}& 1.400\text{lb}& 700\text{lb}\end{array}\right]\begin{array}{c}\text{Alloy}1\\ \text{Alloy}2\end{array}\end{array}$

$\begin{array}{l}\begin{array}{cc}\text{Supplier}A& \text{Supplier}B\end{array}\\ N=\left[\begin{array}{cc}\text{\$0}\text{.75}& \text{\$0}\text{.70}\\ \text{\$6}\text{.50}& \text{\$6}\text{.70}\\ \text{\$0}\text{.40}& \text{\$0}\text{.50}\end{array}\right]\begin{array}{c}\text{Copper}\\ \text{Tin}\\ \text{Zinc}\end{array}\end{array}$

(A) Discuss possible interpretations of the elements in the matrix products $MN\text{ and }NM$.

(B) If either product $MN\text{ or }NM$ has a meaningful interpretation, find the product and label its rows and columns.

(C) Discuss methods of matrix multiplication that can be used to determine the supplier that will provide the necessary materials at the lowest cost.