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### Matrix Multiplication

```
Date: 09/04/97 at 21:25:28
From: Lindz Shank
Subject: Matrix Multiplication

I am doing a project in Algebra 2 and we are broken into groups.
We must research the following topic:

MATRIX MULTIPLICATION

We can use the Internet, reference books, etc...
We must find documented proof of reference.  Can you offer any help?
It would be appreciated.

Thank you.
```

```
Date: 09/15/97 at 12:05:27
From: Doctor Guy
Subject: Re: Matrix Multiplication

Dear Lindz,

I have a book called "Historical Topics for the Mathematics Classroom"
(1969, 1989) by the National Council of Teachers of Mathematics (NCTM)
and it states there that the person who is generally credited with
first inventing matrices was Arthur Cayley in the 1840's and 1850's,
building on work on determinants for solving linear equations that was
done by Cramer, Leibniz, and others.

I have another book called "A History of Mathematics" (2nd edition)
(Boyer & Merzbach), that goes into more historical detail,
pointing out also the contribution of Sylvester, who ended up teaching
at Johns Hopkins.

In any case, a matrix is a rectangular array of numbers, as you
probably know, with the numbers arranged in rows and columns.
Each row of a given matrix must have the same number of elements;
so must each column. Every time someone makes a list of numbers, or,
better yet, a table of numbers, concerning anything at all (prices,
students' grades, populations, coordinates of points, ingredients and
food values, production tables), it can be considered to be a matrix.

Here's an example: sales of tires at various stores.

In the U.S. at least, there are three different sizes of wheels that
rubber auto tires fit on: 13", 14", and 15". There are lots of other
differences between tires, but let's keep it really simple and have
just two more categories: all-weather tires and mud-snow tires.
Let us suppose that at Tires-R-Us this week, they sold 16 of the
13" mud-snow tires, 8 of the 14" mud-snow tires, and... wait a second,
this is going to take too much typing. I will simply make a table and
then you can see it easily for yourself. (At least you can see one of
the good points of matrices, I hope.)

all-weather   mud-snow

13"   |      12           16    |
14"   |       9            8    |
15"   |      22            3    |

It's a bit hard to type this correctly on the internet; the '|' marks
should sort of 'bend' at the ends so that you sort of get a cage, so
that the entire matrix would look like a very big '[ ]' with a lot of
numbers in the middle. I hope you get the idea.

This is known as a 3-by-2 matrix: it has 3 Rows and 2 Columns.
The headings are not part of the matrix itself. A 2-by-3 matrix would
have 2 rows and 3 columns. A horizontal list of 40 items could be
considered to be a 1x40 matrix; a vertical list of 5 items could be
considered to be a 5x1 matrix. A matrix with the same number of rows
and columns is called a square matrix.

If we call the matrix above [T] (for tires), then we can do things
with it. For instance, we could call up the manager of the store and
tell her, "If you don't sell three times as many tires next week in
every single category, then you're fired!" What that means is that we
want her to multiply matrix [T] by 3, or else. The result is exactly
what you expect.

|  36      48  |
3*[T] =   |  27      24  |
|  66       9  |

You can also add matrices, say, from two stores, only if the two
stores keep their matrices the same way. In that case, the element in
row 1, column 1 (the 12, representing the number of 13" all-weather
tires) would be added to the corresponding element in the other
store's matrix, to get the total number of 13" all-weather tires sold.

matrix multiplication. I will attempt to explain how to multiply two
matrices times each other, which is different; previously I showed you
multiplying a matrix times a real number (known as a scalar). By the
way, 3*[T] = [T]*3; it's commutative.

Matrix multiplication is defined in a rather peculiar fashion. If you
want to multiply matrix [A] times matrix [B], in that order, then the
number of columns in matrix [A] must equal the number of rows in
matrix [B]. And it is not commutative!

What you do is multiply the first element in the first row of matrix
[A] by the first element in the first column of [B]; to that product
you add the product of the second element in the first row of matrix
[A] times the second element in the first column of [B]; to that you
add (if it exists) the product of the third element in the first row
of matrix [A] times the third element of the first column of [B]; and
so on to the end of the first row of [A] and the first column of [B].
This sum all goes in the first column, first row of the answer matrix.
The answer matrix has the same number of rows as [A] and the same
number of columns as [B].

I bet that was confusing. There is no way around it, and I'm not
nearly finished explaining how to do it.
| 2   -3 |
An example may be more helpful. Suppose [A] = | 0    6 |
| .1  -9 |
and [B] = |0    8  1|
|.4  -2  7|.

[A] is 3x2 and [B] is 2x3; the result of [A]*[B] will be 3x3; the
result of [B]*[A] will be 2x2. In fact, [B]*[A] can be obtained in
this way:

| (0*2 + 8*0 + 1*.1)   (0*-3 + 8*6 + 1*-9)   |
| (.4*2 + -2*0 + 7*.1) (.4*-3 + -2*6 + 7*-9) |

which I hope equals  |  .1   39    |
|  1.5  -76.2 |  (please check)

but if you multiply [A]*[B] you get

| (2*0 + -3*.4)  (2*8 + -3*-2)  (2*1 + -3*7)  |
| (0*0 + 6*.4)   (0*8 + 6*-2)   (0*1 + 6*7)   |
| (.1*0 + -9*.4) (.1*8 +-9*-2)  (.1*1 + -9*7) |

which I hope equals
| -1.2    22    -19   |
|  2.4   -12     42   |
| -3.6    18.8  -62.9 |  (again, please check)

I hope this helps. There is a lot more I could add, but this is
already quite long.  You can also find a great deal of relevant
material in the Math Forum's Internet Mathematics Library located at:

http://mathforum.org/library/

The Matrices section is at:

http://mathforum.org/library/browse/static/topic/matrices.html

-Doctor Guy,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Definitions
College Linear Algebra
High School Definitions
High School Linear Algebra

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