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## Algebra (all content)

### Course: Algebra (all content) > Unit 20

Lesson 17: Model real-world situations with matrices# Matrix word problem: prices

Sal shows how matrices can be used to efficiently represent data about the prices of toilet paper and toothpaste in two different cities. Created by Sal Khan.

## Want to join the conversation?

- what's the difference between a matrix and just an ordinary chart?(4 votes)
- I'm not sure what you mean by "basic chart" and I suspect that to be the reason why noone has answered your question yet.

If, by basic chart, you mean some type of representation of some multi-dimensional data, then there is really no difference. - If you can take some data and arrange it in rows and columns, then you would certainly have a matrix-representation of that data.(17 votes)

- If matrices are used to represent data for a computer, are matrices a relatively new concept or did they exist even before computers were invented?(6 votes)
- Matrix math has been around for a very long time -- around 3000 years or so. Though, of course, the ancient uses of them was not in the form we use today.(4 votes)

- I see there are two places where matrices are in khanacademy: Precalculus and Algebra 2. Are these the exact same videos and exercises or is there a difference in application (How they are applied to the video/real world) for the two.(3 votes)
- There is a big difference. If you are comfortable with the videos in Pre Calc, then the ones in Alg II will seem very easy as they are more "get to know me" than the Pre calc, which are "how to use me."(5 votes)

- wasn't the third option right?(1 vote)
- No, G(2,1) refers to the second row and first column of the G matrix which is $1.95 not $5.25.(7 votes)

- how do matrices have anything to do with linear equations?(3 votes)
- matrices have things to do with linear equations ,A matrix is a another way to visualize linear equations in a effective way so that the x,y,z..n of the linear equations can be found out simplay(2 votes)

- why cant we just solve the matrix as an equation?(3 votes)
- The whole point of matrices is that we can use them to solve systems that have many equations. So rather than solving one by one, there are operations that quickly reduce a system of equations that are in matrix form into a solution. You will be getting to this soon!(1 vote)

- In the first statement, why coudn't a city have been represented as a diagonal line (a cross) through the matrix? Is it an obligation to only use rows and columns to represent data?(2 votes)
- Then would your first city only be allowed to sell toilet paper?

Diagonals can't work when looking at a fixed number of categories, because you need to fill all the "boxes" or entries for each row and column- a vertical/horizontal grid layout is the only one that works.

You've started me thinking about three dimensional matrixes (is that how you should spell it?) now.(2 votes)

- Does the apostrophe next to the G on the changed matrix represent an alteration on the original matrix?(1 vote)
- This is just a nametag for the matrix.

-----This capital G apostrophe is (pardon my phonetics) read as, "gee prime." In calculus notation this means the function is the first derivative of G. As you might already know, this is useful, because the derivative of a function is the rate of change of a function. (If not, don't worry too much before checking out some videos on intuitive understanding of derivatives.)

-----The G' matrix may be showing that the price changed, or could be used to calculate the rate of change and would be flagged in this way by the owner/author of the matrix&name.(4 votes)

- Instead of matrices, why can't we just use tables?(1 vote)
- This is just an introduction to the concept of a matrix and how it can be used. In the video the appearance of the matrix as a table is intentional to show some similarities as far as organizing data is concerned. As you progress you will understand the power and flexibility of matrices.(3 votes)

- How do you define what rows and columns represent?(2 votes)
- You compare the entries to the information given in the exercice and link columns and rows to a category of information. You can define what rows and columns represent like this.(1 vote)

## Video transcript

Voiceover:The price of
things at two supermarkets are different in different cities. Toilet paper in Duluth,
Minnesota cost 3.99 a package while toilet paper in New
York City cost 8.95 a package. In Duluth, toothpaste costs $1.95 a tube while in New York City
it costs $5.25 a tube. The data for this can be encoded in the following grocery matrix. Let's see if this makes sense. They have the price of
toilet paper in Duluth 3.99 that's this right over here. Then they have the price of toilet paper in New York at 8.95 so that's this right over here. This first row right over here, this is the toilet paper row. I'll just write TP for short,
that's the toilet paper row and this first column
looks like it's Duluth while this second column
looks like it's New York City. Let's see if that works
out for the other data. In Duluth a toothpaste cost $1.95 a tube, so this data right over
here, is right over here. This looks like the toothpaste row. Toothpaste actually is also TP so I'll just write out tooth. This is the toothpaste row, this is toothpaste in Duluth, $1.95. Toothpaste in New York City, 5.25. That's how they have set-up
the data right over here. Which statements are true
about the above matrix? Select all that apply. The following matrix can also be used to contain the same information as G. That's what's interesting
about the matrix, what we have right over here is essentially an encapsulation
of all of the data that we have in this upper paragraph and it's useful because a computer could make use of this data as long as it knows what these rows and what these columns actually represent. The first thing they ask us is whether the following
matrix can also be used to contain the same information. Let's see, this is the price
of toilet paper in Duluth and this right over here is the price of toilet paper in New York City. They have the exact same first row, I would say that looks
like the toilet paper row and that would be Duluth column and this would be the
New York City column. Then if we define things that way then this would have to
be the toothpaste row, but the price of toothpaste in Duluth is not 5.25, and the price of
toothpaste in New York City is not 1.95. This one is not representing
the same information as that up there. You can't just randomly order this around. Now you could represent it in other ways, you could have another ... Let's say I have the matrix A, we could have picked to
do something like this where we could have said well maybe this is the toothpaste column and maybe this is the toilet paper column, and that this first row is New York City and the second row is Duluth. You could have done something like this toothpaste in New York City was 5.25, toilet paper in New York City is 8.95, toothpaste in Duluth is 1.95 and toilet paper in Duluth is 3.99. This would have been, this matrix A that I've just constructed, this does contain the same data because if we appropriately
define our columns and rows to represent toothpaste or toilet paper in New York City or Duluth, you can contain that information. The problem with this one
is it's not consistent. This first row makes us think, okay this must be the Duluth column, this must be the New York City column and that the second row must be toothpaste since the first row is toilet paper but then they got things mixed up. If they switch these two things around then obviously you get the same matrix but then at least the data
would have been consistent. I definitely would not
say that this matrix contains the same information. The second column represents
the price of toothpaste in the two cities. The second column is this right over here, no, that's not the price of
toothpaste in the two cities, that's the price of
the two different goods in New York City, so I wouldn't check that one either. Element G two comma one is equal to 5.25, so when someone says ... If you have the matrix and they give you two comma one. These are the indices of
the row and the column, so this is a second row, first column. Let's go there, second
row so this is one, two, second row, first column
is right over there. G two comma one is not 5.25, so I wouldn't check that one either. A change in the price of
toilet paper in Duluth can be represented by the matrix. Toilet paper in Duluth is this entry of our matrix right over there. Change in toilet paper in Duluth would represent a change in this 3.99 but they didn't change the 3.99, they changed the 9.75. The 9.75 would represent a change in the price of toilet
paper in New York City. I wouldn't check that one either, so I would only go with none of the above.