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Date: 24 Feb 1995 18:58:12 -0500
From: Steve Toub
Subject: A matrix and its inverse

Dear Dr. Math-

In using my calculator to figure out matrices, I use the formula,
|A|^-1|B|.  I know that I am multiplying the inverse of matrix |A| by
the matrix |B|, but I don't UNDERSTAND why I am doing this.
What is the inverse of a matrix?  How would you find this?  How do
you multiply two matrices together?  I am fairly knowledgeable in
mathematics, but could you try to explain this so that a pre-calculus
student can understand.

Thanks,
Steve
```

```
Date: 27 Feb 1995 19:28:40 -0500
From: Dr. Ken
Subject: Re: A matrix and its inverse

Hello there!

Well, here's the story.  Remember back when you were learning about
multiplication of numbers?  You learned that if you multiply any number
by 1, you'll get that number.  And you learned that if you multiply 6 by 1/6,
or 3/4 by 4/3, you'll get 1.  Well, that's essentially the situation that's
happening with the matrices.

You may know that when you multiply any n-by-n matrix by the identity
matrix (the n-by-n matrix that has all ones down the main diagonal and
zeroes everywhere else), you get that same matrix back again.  So if we
let the letter, I, represent the n-by-n identity matrix, and A represent
any other n-by-n matrix, then we have A * I = A and I * A = A.  This is
much like the situation in the real numbers:  x * 1 = x and 1 * x = x.

Where do inverses come in?  Well, the inverse of an n-by-n matrix, A, is
the matrix you can multiply A by to get I, just like the inverse (reciprocal)
of a real number, x is the number you can multiply x by (1/x) to get 1.

Remember that for a second.  Now I'll see if I can explain matrix
multiplication to you.  Let's say we have two matrices A and B:

1  2  2        3  4  5
4  7  6        5  9  9
5  6  0        2  7  3

A              B

To multiply them together, we take a row from A and a column from B and
multiply each entry in the row and the column together, and then add up what
we get.  The result will be a new 3x3 matrix:

1*3 + 2*5 + 2*2   1*4 + 2*9 + 2*7   1*5 + 2*9 + 2*3      17  36  29
4*3 + 7*5 + 6*2   4*4 + 7*9 + 6*7   4*5 + 7*9 + 6*3  =   59 121 101
5*3 + 6*5 + 0*2   5*4 + 6*9 + 0*7   5*5 + 6*9 + 0*3      45  74  79

A*B                                         A*B

Notice that multiplication of matrices isn't commutative: if you multiply
B*A you'll get something completely different, so A*B doesn't equal B*A.
Also notice that you can't multiply a 3x4 matrix by a 5x2 matrix.  If you
have a something-by-n matrix, you can only multiply it on the right by an
n-by-something-else matrix.  By the way, I think multiplication of matrices
is something you'd understand a lot better if you got someone to actually
show it to you, instead of trying to just tell you like I did.

Now back to the question of inverses.  Here is an example of a matrix and
its inverse (in fact, it's A and its inverse):

1  2  2        -18    6   -1             1  0  0
4  7  6         15   -5    1             0  1  0
5  6  0         -5.5  2   -.5            0  0  1

A              A Inverse           A * A Inverse

Why would we ever want to study inverses?  For a lot of the reasons we
studied reciprocals.  If we had the equation x * y = 5, then we could
mutiply both sides on the left by 1/x, and we'd get y = 5/x.  Well, it's
the same with matrices: if we have the equation A * B = C, we can
multiply by A^-1 and we'll get B = C * A^-1.

So that's what inverses are.  Keep in mind that this is a tricky subject:
the whole college course Linear Algebra is essentially the study of
matrices.  So don't give up, and try experimenting with them some
more!
```

```
Date: 27 Feb 1995 21:54:27 -0500
From: Steve Toub
Subject: Re: Re: A matrix and its inverse

Ken-

Thank you so much for your help with matrices.  I totally understand the
multiplication part now, and I understand the idea of an inverse and what
it is, but not how to find it.  Is there some sort of formula or equation to
finding an inverse?

The reason that I am asking these questions is that I am in the process
of writing a shareware math stack for High School aged kids.  The
programming part is easy, it is the math that I have a little knowledge
problem with.

By the way, if I ever get the program done, I will be sure to include your
name under the credits.

Thanks,
Steve
```

```
Date: 3 Mar 1995 12:37:19 -0500
From: Dr. Ethan
Subject: Re: Re: A matrix and its inverse

Hey, this isn't Ken but I should be able to help you.

Do you know how to row reduce a matrix?  That is the process
that we will be using.  In row reducing there are three things that you
can legally do:

1. You can switch two rows.
2. Multiply a row by a constant
3. Add one row plus a constant times another and replace one by the sum.

Let's see an example.

| 2 4 3 | = r1      So we could use    r3 | 1 1 0 |
| 1 3 2 | = r2      move one to get    r2 | 1 3 2 |
| 1 1 0 | = r3                         r1 | 2 4 3 |

Move three   r3 | 1 1 0 |
r2 - r3 | 0 2 2 |
r1 - 2r3 | 0 2 3 |

Now we get to how to find inverses.

If we want the inverse of the matrix that we have been dealing with
we write it like this

| 2 4 3 | 1 0 0 |
| 1 3 2 | 0 1 0 |
| 1 1 0 | 0 0 1 |

Then we use the techniques of row reduction to get the left side to
look like the right side.  Whatever remains on the right side is the
inverse.  Let's see how it works.  We'll start with the steps I

r3 | 1 1 0 | 0 0 1 |             r3 | 1 1 0 | 0 0  1 | relabel a1
r2 | 1 3 2 | 0 1 0 |  then  r2 - r3 | 0 2 2 | 0 1 -1 |   a2
r1 | 2 4 3 | 1 0 0 |       r1 - 2r3 | 0 2 3 | 1 0 -2 |   a3

next a1 - .5 a2 | 1 0 -1 | 0 -.5  1.5 | relabel w1
.5 a2 | 0 1  1 | 0  .5  -.5 |   w2
a3 - a2 | 0 0  1 | 1  -1   -1 |   w3

then  w1 +  w3 | 1 0 0 |  1 -1.5  .5 |
w2 -  w3 | 0 1 0 | -1  -.5  .5 |
w3 | 0 0 1 |  1   -1  -1 |

Okay, so now the matrix on the right should be the inverse.
You can check this by multiplying it by the original matrix using
the method taught to you by Ken.

There is another method for finding inverses involving things
called determinants.  However I think that it would be most
useful for you to get a linear algebra book to see it explained
as it is long and involves more linear algebra.
Hope that helps.

Ethan Doctor On Call
```
Associated Topics:
High School Linear Algebra

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