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### 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

(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/
```
Associated Topics:
High School Linear Algebra

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