The problem describes a shopping complex with three areas: food (F), clothing (C), and merchandise (M). We are given the proportion of shoppers in each area by matrix $P$ and the average daily amount spent in each area in 2020 by matrix $A_{2020}$. The problem has three parts: a) Determine the order of matrix $P$. b) Calculate the total number of shoppers if there are 800 people in the clothing area. c) Show that the total amount spent by 135 food shoppers, 143 clothing shoppers, and 131 merchandise shoppers is $9663.20, using the matrix $A_{2020}$.

1. Problem Description

The problem describes a shopping complex with three areas: food (F), clothing (C), and merchandise (M). We are given the proportion of shoppers in each area by matrix $P$ and the average daily amount spent in each area in 2020 by matrix $A_{2020}$. The problem has three parts:

a) Determine the order of matrix $P$.

b) Calculate the total number of shoppers if there are 800 people in the clothing area.

c) Show that the total amount spent by 135 food shoppers, 143 clothing shoppers, and 131 merchandise shoppers is $9663.20, using the matrix$A_{2020}$.

2. Solution Steps

a) The order of a matrix is given by (number of rows) x (number of columns). The matrix $P$ is given by

$P = \begin{bmatrix} 0.48 \\ 0.27 \\ 0.25 \end{bmatrix}$.

This matrix has 3 rows and 1 column. Therefore, the order of the matrix $P$ is 3x

1.

b) Let $T$ be the total number of shoppers. We are given that the proportion of shoppers in the clothing area is 0.

2

7. Therefore, $0.27 \times T =$ number of shoppers in the clothing area.

We are given that the number of shoppers in the clothing area is

8

0

0. So, we have $0.27T = 800$.

To find the total number of shoppers $T$, we divide both sides of the equation by 0.27:

$T = \frac{800}{0.27} = 2962.96$

Since the number of shoppers must be a whole number, we can round this to

2

9

6

3.

c) We are given the average daily amount spent by each shopper in each area by matrix $A_{2020}$:

$A_{2020} = \begin{bmatrix} 21.30 \\ 34.00 \\ 14.70 \end{bmatrix}$

We are also given the number of shoppers in each area: 135 food shoppers, 143 clothing shoppers, and 131 merchandise shoppers. We can represent this as a matrix $S$:

$S = \begin{bmatrix} 135 \\ 143 \\ 131 \end{bmatrix}$

Let the total amount spent be $T$.

$T =$ (number of food shoppers) $\times$ (average amount spent on food) + (number of clothing shoppers) $\times$ (average amount spent on clothing) + (number of merchandise shoppers) $\times$ (average amount spent on merchandise).

$T = (135 \times 21.30) + (143 \times 34.00) + (131 \times 14.70)$

$T = 2875.50 + 4862.00 + 1925.70$

$T = 9663.20$

We can represent this as a matrix multiplication. Let $S' = \begin{bmatrix} 135 & 143 & 131 \end{bmatrix}$. The matrix $S'$ is obtained by transposing S.

Then, we need to express how many shoppers per category. In terms of matrices

Total = $\begin{bmatrix} 135 & 143 & 131 \end{bmatrix} \times A_{2020}$. Note that $A_{2020}$ must be in the form where the categories are aligned.

This problem setup is a bit confusing.

However, it is known that each shopper spent a certain amount.

So, in vector form, the amount of spent equals $\begin{bmatrix} 135 & 143 & 131 \end{bmatrix} \begin{bmatrix} 21.30 \\ 34.00 \\ 14.70 \end{bmatrix} = 135(21.30) + 143(34.00) + 131(14.70) = 2875.5+4862+1925.7=9663.20$.

3. Final Answer

a) 3x1

b) 2963

c) $\begin{bmatrix} 135 & 143 & 131 \end{bmatrix} \begin{bmatrix} 21.30 \\ 34.00 \\ 14.70 \end{bmatrix} = 9663.20$