Question

as shoes and bs footwear both have outlets in city 1 and city 2. as sells shoes for $90, sandals for $60, and boots for $140. bs prices are $40, $30, and $170 for shoes, sandals, and boots, respectively. half of all sales in city 1 stores are shoes, 1/4 are sandals, and 1/4 are boots. in city 2 the fractions are 1/5 shoes, 1/5 sandals, and 3/5 boots. complete parts (a) through (d) below.
(a) write a 2×3 matrix called p representing prices for the two stores and three types of footwear.
p = 90 60 14040 30 170 (type an integer or simplified fraction for each matrix element.)
(b) write a 3×2 matrix called f representing the fraction of each type of footwear sold in each city.
f = 1/2 1/51/4 1/51/4 3/5 (type an integer or simplified fraction for each matrix element.)
(c) only one of the two products pf and fp is meaningful. determine which one it is and calculate the product. select the correct choice below and fill in the answer box to complete your choice. (type an integer or simplified fraction for each matrix element.)
a. the product pf = is meaningful.
b. the product fp = is meaningful.

Explanation:

Step1: Recall matrix - multiplication rules

For two matrices $A_{m\times n}$ and $B_{n\times p}$, the product $AB$ is defined and is a $m\times p$ matrix. The product $BA$ is defined if and only if $p = m$ and is a $n\times n$ matrix. Matrix $P$ is a $2\times3$ matrix and matrix $F$ is a $3\times2$ matrix.

Step2: Check the product $PF$

For the product $PF$, the number of columns in $P$ (which is 3) is equal to the number of rows in $F$ (which is 3). So, the product $PF$ is a $2\times2$ matrix.

Step3: Calculate the product $PF$

The $(i, j)$ - entry of the product matrix $PF$ is given by the dot - product of the $i$ - th row of $P$ and the $j$ - th column of $F$.
For the first row of $P$ and first column of $F$:

$$90\times\frac{1}{2}+60\times\frac{1}{4}+140\times\frac{1}{4}=45 + 15+35=95$$

For the first row of $P$ and second column of $F$:

$$90\times\frac{1}{5}+60\times\frac{1}{5}+140\times\frac{3}{5}=\frac{90 + 60+420}{5}=\frac{570}{5}=114$$

For the second row of $P$ and first column of $F$:

$$40\times\frac{1}{2}+30\times\frac{1}{4}+170\times\frac{1}{4}=20+\frac{30 + 170}{4}=20 + 50=70$$

For the second row of $P$ and second column of $F$:

$$40\times\frac{1}{5}+30\times\frac{1}{5}+170\times\frac{3}{5}=\frac{40 + 30+510}{5}=\frac{580}{5}=116$$

So, $PF=

$$\begin{bmatrix}95&114\\70&116\end{bmatrix}$$

$

Answer:

A. The product $PF=

$$\begin{bmatrix}95&114\\70&116\end{bmatrix}$$

$ is meaningful.