Matrix Multiplication

Date: 10/11/2003 at 18:11:26
From: Brooke
Subject: multiplying matrices

I have been trying to do this problem for a while, but I'm not getting 
the correct answer.  Can you help?

  [6 4 1] x    2  5
            [ -3  0  ]
              -1  3 

I think the result has to have dimensions 1x2, but I don't know how to 
do the multiplication. 

Date: 10/24/2003 at 01:19:46
From: Doctor Ian
Subject: Re: multiplying matrices


Hi Brooke, 

Let's look at two matrices like

   | a b |  | u v w |
   | c d |  | x y z |
   | e f |

The product of the matrices will look like

   | a b |   | u v w |   | ? ? ? |
   | c d | * | x y z | = | ? ? ? |
   | e f |               | ? ? ? |

Each '? will be replaced by the dot product of a row of the first 
matrix, and a column of the second.  For example, 

   | a b |   | u v w |   | ? ? ? |
   | c d | * | x y z | = | ? ? ? |
   | e f |               | ? _ ? |
                             ^
                             |
                     element in 3rd row, 2nd column
                     is dot product of 3rd row of first matrix
                     and 2nd column of second matrix

Note that this is _why_ the number of columns in the second matrix has
to be the same as the number of rows in the first.

What is a dot product?  Note that each row of the first matrix has the
same number of elements as each column of the second matrix.  This
isn't an accident.  It's a requirement.  To find the dot product of
the row (e,f) and the column (v,y), we multiply the corresponding
elements and add the products:

  (e,f).(v,y) = ev + fy

So this is what goes in the result matrix:

   | a b |   | u v w |   | ?     ?     ? |
   | c d | * | x y z | = | ?     ?     ? |
   | e f |               | ?   ev+fy   ? |

Is this enough to get started? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/