There are three convenience stores in Folsom. This week,...

There are three convenience stores in Folsom. This week, store I sold 88 loaves of bread, 48 qt of milk, 16 jars of peanut butter, and 112 lb of cold cuts. Store II sold 105 loaves of bread, 72 qt of milk, 21 jars of peanut butter, and 147 lb of cold cuts. Store III sold 60 loaves of bread, 40 qt of milk, no peanut butter, and 50 lb of cold cuts. a. Use a $4 \times 3$ matrix to express the sales information for the three stores. b. During the following week, sales on these products at store I increased by 25$\%$ ; sales at store II increased by $1 / 3 ;$ and sales at store III increased by 10$\% .$ Write the sales matrix for that week. c. Write a matrix that represents total sales over the two-week period.

There are three convenience stores in Folsom. This week, store I sold 88 loaves of bread, 48 qt of milk, 16 jars of peanut butter, and 112 lb of cold cuts. Store II sold 105 loaves of bread, 72 qt of milk, 21 jars of peanut butter, and 147 lb of cold cuts. Store III sold 60 loaves of bread, 40 qt of milk, no peanut butter, and 50 lb of cold cuts.
a. Use a $4 \times 3$ matrix to express the sales information for the three stores.
b. During the following week, sales on these products at store I increased by 25$\%$ ; sales at store II increased by $1 / 3 ;$ and sales at store III increased by 10$\% .$ Write the sales matrix for that week.
c. Write a matrix that represents total sales over the two-week period.

Key Concepts

Matrix Representation of Data

This concept involves organizing related numerical information into a structured array of rows and columns. It is used to compactly represent data sets like sales figures, inventory numbers, or any other measurements in a way that facilitates further mathematical operations, analysis, and interpretation.

Scalar Multiplication

Scalar multiplication refers to the technique of multiplying every entry in a matrix by a constant number (a scalar). This is particularly useful when scaling data, such as increasing or decreasing sales figures by a given percentage, to reflect changes over time or in response to varying conditions.

Matrix Addition

Matrix addition is the process of adding corresponding entries from two matrices to form a new matrix. This operation is essential for combining data from different sources or time periods, such as aggregating sales information over multiple weeks, ensuring that the overall totals are correctly computed.

Data Transformation in Matrices

Data transformation in matrices involves applying various operations such as scaling, adding constants, or combining matrices to change or update the original dataset. This helps in modeling changes in real-world situations, like percentage increases or decreases in sales, and enables comparative analysis across different time intervals or conditions.

Transcript

00:01 Okay, so in this question and part a, we have to come up with a 4x3 matrix that can handle 4 items and 3 stores.

00:10 So we know what the rows are going to correspond to and we know what the columns are going to correspond to because there are four items and there are three stores.

00:18 Each row must and it's a 4 by 3 matrix.

00:21 So remember this corresponds to the number of rows.

00:24 So the first one corresponds to the number of rows and the second one corresponds to the columns, right? so if it's four by three, four rows and three columns, and we have four items and we have three stores.

00:37 So each row represents an item and each column represents a store.

00:43 So we're given this information in the question and now we just need to fill it in.

00:46 So the first row will correspond to bread.

00:48 The second row corresponds to milk.

00:50 The third row corresponds to peanut butter.

00:52 And the fourth row corresponds to cold cuts.

00:54 And the first column corresponds to store.

00:57 Second column corresponds to the second store.

00:59 And the third column corresponds to the third store.

01:01 So in the question we know, the number of bread in store 1 is 88.

01:05 So row 1, column 1 is 88.

01:09 So let's fill this out column by column because that's how it's given in the question.

01:13 Now in store 1, there's 48 quarts of milk.

01:16 So the second row corresponds to 48.

01:18 So second row first column is 48.

01:20 Now there's 16 jars of peanut butter in store 1.

01:23 So the third row is peanut butter, and the first column is store 1.

01:27 At 16 here and similarly 112 will be the fourth row in first column now similarly in store three so store 2 is going to be filled let's just do store 3 in store 3 there's 60 lows of bread so the third column first row corresponds to 60 second row third column is the number of milk in store 3 which is 40 there's no jars of peanut butter hence there's a zero here and 50 a cold 50 pounds of cold cut which is basically the 4 fourth row and third column.

01:59 So it's just, this is just entering the information given in the question in the matrix, in a matrix form.

02:05 Okay, so now let's go to part b.

02:07 Now, in part b, they tell you that in store one, the sales have increased by 25%.

02:12 An increase of 25 % means basically you need to multiply by 1 .25.

02:20 Now, the 0 .25 corresponds to the 25 % and the 1 is an increase, right? for example, let's just do the second one.

02:30 So the quartz of milk sold in week one and store one was 48.

02:34 So now there's going to be an increase of 25%.

02:37 Now, 25 % corresponds to one -fourth.

02:39 Remember this, right? because 25 % is 25 over 100, which is 1 over 4.

02:45 So an increase of 1 over 4 and 48 is 12.

02:48 But now you need to add this to the original amount that because you sold an increase of 25%.

02:55 So that's going to be 48.

02:57 You sold originally, and the increase is 12.

02:59 So it's 60 total.

03:01 So basically every element of this first column needs to be multiplied by 1 .25 because everything sold increased by a 25%.

03:10 So you take the 25 % of this and add it to itself.

03:14 So that's 1 .25 times every element of column 1.

03:19 And you will get these answers.

03:22 So let's just do another one.

03:23 So 25 % of 16 is 4.

03:26 4 plus 16 is 20.

03:28 So the answers are right.

03:30 Now let's go to, so this is just a good way of always checking whether you have the right answer or not.

03:38 Just pick one and see whether you're getting the right answer.

03:41 So the easiest computation here was with the 16 one.

03:44 Now let's go to the second store.

03:46 Now in the second store, there's an increase of one third for each item...

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