Shoe Sales Sal's Shoes and Fred's Footwear both have outlets...

Shoe Sales Sal's Shoes and Fred's Footwear both have outlets in California and Arizona. Sal's sells shoes for $\$ 80,$ sandals for $\$ 40,$ and boots for $\$ 120$ . Fred's prices are $\$ 60, \$ 30,$ and $\$ 150$ for shoes, sandals, and boots, respectively. Half of all sales in California stores are shoes, 1$/ 4$ are sandals, and 1$/ 4$ are boots. In Arizona the fractions are 1$/ 5$ shoes, 1$/ 5$ sandals, and 3$/ 5$ boots. a. Write a $2 \times 3$ matrix called $P$ representing prices for the two stores and three types of footwear. b. Write a $3 \times 2$ matrix called $F$ representing the fraction of each type of footwear sold in each state. c. Only one of the two products $P F$ and $F P$ is meaningful. Determine which one it is, calculate the product, and describe what the entries represent. d. From your answer to part $c,$ what is the average price for a pair of footwear at an outlet of Fred's in Arizona?

Shoe Sales Sal's Shoes and Fred's Footwear both have outlets in California and Arizona. Sal's sells shoes for $\$ 80,$ sandals for $\$ 40,$ and boots for $\$ 120$ . Fred's prices are $\$ 60, \$ 30,$ and $\$ 150$ for shoes, sandals, and boots, respectively. Half of all sales in California stores are shoes, 1$/ 4$ are sandals, and 1$/ 4$ are boots. In Arizona the fractions are 1$/ 5$ shoes, 1$/ 5$ sandals, and 3$/ 5$ boots.
a. Write a $2 \times 3$ matrix called $P$ representing prices for the two stores and three types of footwear.
b. Write a $3 \times 2$ matrix called $F$ representing the fraction of each type of footwear sold in each state.
c. Only one of the two products $P F$ and $F P$ is meaningful. Determine which one it is, calculate the product, and describe what the entries represent.
d. From your answer to part $c,$ what is the average price for a pair of footwear at an outlet of Fred's in Arizona?

Key Concepts

Matrix Representation

This concept involves organizing data into rows and columns to form a matrix. In many applications, such as representing prices or proportions, matrices provide a structured way to store and manipulate information, making it easier to apply algebraic operations on the data.

Matrix Multiplication and Dimensional Compatibility

Matrix multiplication is an operation that combines two matrices by taking the dot product of rows of the first matrix with the columns of the second. A crucial aspect is that the number of columns in the first matrix must match the number of rows in the second. This concept ensures that only products with matching inner dimensions are defined and meaningful.

Non-Commutative Nature of Matrix Multiplication

Unlike scalar multiplication, matrix multiplication is generally non-commutative, meaning that the order in which you multiply matrices matters. One product might yield a meaningful result while reversing the order of the factors could result in an undefined or contextually irrelevant matrix, highlighting the importance of the correct multiplication order.

Interpreting Matrix Products in Context

When matrices represent real-world data, their product can be used to compute weighted sums or averages. Each entry in the resulting matrix is the sum of products representing a weighted combination of the original data values, such as computing average prices based on sale fractions, thus providing insights into aggregated outcomes.

Transcript

00:01 For this problem, we are looking at two footwear stores, sal's shoes and fred's footwear.

00:06 Each of them have outlets in two states and sell three different types of shoes at different prices.

00:12 So we're going to create a few matrices to help us track all of this information.

00:16 First, we are going to make a matrix called p.

00:20 P is going to be a two by three matrix, so two rows, three columns.

00:26 And we want to see all of the prices for all of the footwear that these shoes.

00:31 Shoes these shoe stores sell.

00:33 So i'm going to let each of my rows equal one of my shoe stores, sal's and freds.

00:39 And across my columns, those are going to be the three different types of things that they sell.

00:44 They sell shoes, sandals, and boots.

00:54 Okay.

00:55 So let's see what their prices are.

00:58 Sal's sell shoes for $80, sandals are $40, and boots are $120.

01:07 What about first? what about well, his prices are 60, 30, and 150.

01:14 So p tells us all of the prices for all of the different types of shoes that they sell.

01:19 Now, we want to put together another matrix f.

01:24 And we're told f is going to be a 3 by 2.

01:27 So i'll make it a little bit taller.

01:30 3 by 2.

01:31 And here we want to keep track of the fraction of each type of footwear that's sold.

01:36 So in this case, you know, and that is going to be per state.

01:41 So i'm going to need, i've got two states.

01:43 So those are going to need to be my columns.

01:45 I'm going to have arizona and california.

01:49 What i'm tracking in each of those is how many of each type of shoe are sold.

01:56 So those shoe types will go down my rows.

01:59 And i'm going to put them in the same order as they are in the original.

02:06 Shoes, sandals, boots.

02:08 So let's look at california first.

02:11 How are the shoe sales distributed in california? half of all sales are shoes, and then a quarter each for sandals and boots.

02:23 Arizona is broken up a little differently.

02:26 In arizona, we have one -fifth our shoes and sandals, and the remaining three -fifths are boots.

02:34 That's a three there.

02:35 Okay.

02:36 So here are my two matrices, p and f.

02:39 Now, we want to multiply them together.

02:41 One way makes sense.

02:43 One way does not.

02:44 What if i multiply p times f? well, if i did pf, that means i'd be going across the columns and across the row down the columns...

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