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  |

(please check). 

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. 

But you didn't really ask about matrix addition. You asked about 
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/