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Solving Linear Equations

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides sample which develop concepts related to using a matrices to solve linear equations.

I’m sure you remember our whole unit on solving linear equations…by graphing, by substitution, and by elimination. Well, now we’re going to find a new way of solving those equations…by using matrices!

Oh, come on…why do we need another way when we’ve already got three?

Glad you asked! There are two reasons. First, this new method can be done entirely on a calculator. Cool! We like calculators.

Yeah, I know. But here’s the even better reason. Suppose I gave you three equations with three unknowns, and asked you to do that on a calculator. Think you could do it? Um…it would take a while. How about four equations with four unknowns? Please don’t do that. With matrices and your calculator, all of these are just as easy as two. Wow! Do those really come up in real life? Yes, all the time. Actually, this is just about the only “real-life” application I can give you for matrices, although there are also a lot of other ones. But solving many simultaneous equations is incredibly useful. Do you have an example? Oh, look at the time! I have to explain how to do this method.

So, here we go. Let’s start with a problem from an earlier homework assignment. I gave you this matrix equation:

[ 9 -6 18 9 ] [ x y ] = [ 9 -3 ] [ 9 -6 18 9 ][ x y ]=[ 9 -3 ]

Exercise 1

The first thing you had to do was to rewrite this as two equations with two unknowns. Do that now. (Don’t bother solving for x x and y y, just set up the equations.)

The point is that one matrix equation is the same, in this case, as two simultaneous equations. What we’re interested in doing is doing that process in reverse: I give you simultaneous equations, and you turn them into a matrix equation that represents the same thing. Let’s try a few.

Exercise 2

Write a single matrix equation that represents the two equations:

3 x + y = -2 3x+y=-2

6 x - 2 y = 12 6x-2y=12

Exercise 3

Now, let’s look at three equations:

7 a + b + 2 c = -1 7a+b+2c=-1

8 a - 3 b = 12 8a-3b=12

a - b + 6 c = 0 a-b+6c=0

  • a. Write a single matrix equation that represents these three equations.
  • b. Just to make sure it worked, multiply it out and see what three equations you end up with

OK, by now you are convinced that we can take simultaneous linear equations and rewrite them as a single matrix equation. In each case, the matrix equation looks like this:

AX = BAX=B

where AA is a big square matrix, and XX and BB are column matrices. XX is the matrix that we want to solve for—that is, it has all our variables in it, so if we find what XX is, we find what our variables are. (For instance, in that last example, XX was abcabc size 12{ left [ matrix { a {} ## b {} ## c } right ]} {}.) So how do you solve something like this for XX? Time for some matrix algebra! We can’t divide both sides by AA, because we have not defined matrix division. But we can do the next best thing.

Exercise 4

Take the equation AX = BAX=B, where AA, XX, and BB are all matrices. Multiply both sides by A-1A-1 (the inverse of AA) in front. (Why did I say “in front?” Remember that order matters when multiplying matrices. If we put A-1A-1 in front of both sides, we have done the same thing to both sides.)

Exercise 5

Now, we have A-1AA-1A—gee, didn’t that equal something? Oh, yeah…rewrite the equation simplifying that part.

Exercise 6

Now, we’re multiplying II by something…what does that do again? Oh, yeah…rewrite the equation again a bit simpler.

We’re done! We have now solved for the matrix XX.

So, what good is all that again?

Oh, yeah…let’s go back to the beginning. Let’s say I gave you these two equations:

3 x + y = –2 3x+y=–2

6 x 2 y = 12 6x2y=12

You showed in #2 how to rewrite this as one matrix equation AX = BAX=B. And you just found in #6 how to solve such an equation for XX. So go ahead and plug AA and BB into your calculator, and then use the formula to ask your calculator directly for the answer!

Exercise 7

Solve those two equations for xx and yy by using matrices on your calculator.

Did it work? We find out the same way we always have—plug our xx and yy values into the original equations and make sure they work.

Exercise 8

Check your answer to #7.

Exercise 9

Now, solve the three simultaneous equations from #3 on your calculator, and check the answers.

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