# High School: Number and Quantity

### Vector and Matrix Quantities HSN-VM.C.6

6. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships related in a network.

Students should understand what a matrix is and how to use it. No, not The Matrix.

The textbook definition of a matrix is, "A rectangular array of numbers." Yeah, that helps a lot. A more workable definition might be, "An organizational system that sorts numbers into rows and columns."

Students can think of matrices as organizing information into a chart—one column for rate, another for time, and another for distance, for example. That's kind of what we're talking about here.

In a matrix, rows are the horizontal groups of numbers, reading from left to right. Columns are the vertical groups of numbers, reading from top to bottom. Just like the rows and columns in excel spreadsheets and Bingo boards.

Let's say you're preparing your students for college applications. (Good luck.) They can sort out their "safe" schools, "reach" schools, and a number of schools that fall somewhere in between. As they wait for those wonderfully fat envelopes full of good news, you can subtly hint to them that they should figure out finances. Using matrices.

First, they can make a row for each school. Then make columns for tuition, room and board costs, definite scholarships, work-study, and miscellaneous—any other sources of income that might not fit anywhere else.

Their end result might look something like this, but filled with numbers other than 0.

 Schools Tuition Room and Board Scholarships Work-Study Miscellaneous Standford 0 0 0 0 0 Princetown 0 0 0 0 0 Harward 0 0 0 0 0 Yayle 0 0 0 0 0

There, at a glance, they'll have their financial picture for college. No wading through pages and pages of tactful letters; instead, a simple matrix listing what each school will cost them.

Essentially, students should understand that matrices are friends, not enemies.

#### Drills

1. In the above example about college applications, what's the appeal of using a matrix?

All of the above

What can we tell you? Organization is the way to go. That's something students should understand, too. It might take some practice, but matrices are a great way to organize a whole bunch of data.

2. Would it be possible to reorganize the matrix above so that each row represents an expense and each column represents a college?

Yes, as long as the numbers are still organized per college and per expense

Matrices represent an organizational system for numbers, but they don't specify rules for how the numbers need to be organized. Granted, we would need either rows or columns to be types of expenses and the other to be the different colleges so that the matrix makes sense, but it's completely our choice how we want to achieve that.

3. Matrices can also help organize systems of equations or expressions. For example, which of the following matrices represents the expressions 17x + 19y – 2z and 2x – 20y + 23z?

While it doesn't matter which column represents which variable, all number in that column must be coefficients of the same variable. Otherwise, our organization goes out the window. The only choice that doesn't mix up the coefficients between each other is (B). It neatly displays the matrices in terms of the z, y, and then x coefficients. (It's okay that it's "backwards" as long as it displays the right information properly!)

4. Which of the following a matrix representing 8x + 7y = 25 and 9y – 3x = 3?

What might be difficult about this is that the equations are a bit jumbled. While the first equation goes in order of x, then y, and then the constant, the other has y first, and then x. If we know that 8 and -3 are the coefficients for x, all we have to do is make sure they're in the same column. That is only true for (A).

5. Which system of equations does this matrix represent?

7y + 3x = 10, 2y + 9x = 17, and 3y – 2x = 1

The important characteristic of matrices is organization. As long as all the columns match a particular variable and all the rows match a given equation, then we can organize the matrix however we want. In this case, (D) is right because all the numbers in the first column are the coefficients for y, the second column for x, and the final column for their sum. The other three answers mix some x and y values, so they aren't correct.

6. Which matrix might be used to represent the system of equations 3x + 4y = 25 and 2xy = 2?

Answer (B) is wrong because the y coordinates are switched, and (C) is wrong because it doesn't pertain to this particular problem. Even though (D) does represent the correct values for x and y, it isn't a matrix.

7. In the above example, with the system of equations, would it affect the problem if you were to switch the top and bottom rows?

It would change the matrix, but still be appropriate for the problem

If you have a system of equations, it doesn't matter which is written first, as long as you change all the coefficients correctly. That is, every row represents a particular equation and every column a particular coefficient. We can switch entire rows and columns, sure, but not individual values.

8. Given this matrix as a system of equations for ax + by = c, what are the values of x and y?

x = 1, y = 2

We know that each row represents an equation and each column represents the coefficient for a variable. If we add the two rows of the matrix together (which represent the equations 4x + 3y = 10 and -3x – 3y = -9), we get a row of 1, 0, and 1. That means 1x + 0y = 1, or x = 1. Then, we can use algebra to find y. See how useful matrices can be?

9. Given this matrix as a system of equations, what are the values of x and y?

Not enough information is given

The problem here is that we can't be certain which column represents which coefficient. Sure, we could say the first column is x and the second is y, but how can we know? Maybe the matrix-maker meant something different. Matrixes are arranged numbers, but we can't do much with them if we don't know what they stand for.

10. Which of the following represents the system of equations 2x – 3y = 5, 7x + 19y = -12, and x = 1?