Associated Topics || Dr. Math Home || Search Dr. Math

### Why Does Matrix Multiplication Work the Way It Does?

```Date: 09/13/2006 at 22:07:14
From: Casey
Subject: Process of Matrix Multiplication

Can you please explain WHY the process of matrix multiplication
subtracting and scalar multiplying make sense, but multiplication is
complicated and the rationale doesn't make sense to me.

I understand the process.  I just want to know why it works.  Thank you.

```

```
Date: 09/16/2006 at 09:53:49
From: Doctor Fenton
Subject: Re: Process of Matrix Multiplication

Hi Casey,

Thanks for writing to Dr. Math.

Suppose you have a system of linear equations (for simplicity, I will
use two equations in two unknowns, but the principle applies to
systems of all sizes).

a11*x1 + a12*x2 = d1
a21*x1 + a22*x2 = d2   .

(The indices are chosen to indicte the row and column of the
coefficient, so a12 is the coefficient in the first equation of the
second variable, x2 .)

In solving this system by elimination, you can save a lot of writing
by using position, instead of writing all the variable names, plus,
minus, and equals signs, and carrying out row operations on the
augmented matrix

[ a11 a12 : d1 ]
[ a21 a22 : d2 ]

(I have inserted colons to emphasize that the numbers in the last
column are different from the other columns, being the data values on
the right side of the equations, while the other columns are
coefficients of variables.)  In this process, the coefficients become
entries in the coefficient matrix.

Now, suppose we want to make a linear change of variables, so that we
introduce new variables y1 and y2 for which

x1 = b11*y1 + b12*y2
x2 = b21*y1 + b22*y2  .

(Such changes occur if we rotate the coordinate axes, for example,
where the y's denote the coordinates in the rotated system.)

If you substitute these formulas into the original system of
equations, what will the coefficient matrix of the new linear system
of equations in the y-variables be?  For example, the first equation

a11*x1 + a12*x2 = d1

becomes

a11*(b11*y1 + b12*y2)  + a12*(b21*y1 + b22*y2) = d1   ,

or, rearranging,

(a11*b11 + a12*b21)*y1 + (a11*b12 + a21*b22)*y2 = d1 ,

so the first row of the new coefficient matrix for the y-variables is

[ (a11*b11 + a12*b21)  (a11*b12 + a21*b22) ]
[         ...                  ...         ]

If we designate the coefficient matrix for the new system of equations
in the y variables to be the matrix C , where

C = [ c11 c12 ]
[ c21 c22 ] ,

then comparing the entries in the two matrices of coefficients, we see
that

c11 = a11*b11 + a12*b21    and  c12 = a11*b12 + a21*b22  .

After you have written out the full coefficient matrix for the
y-system of equations, compare it with the two coefficient matrices
which are the original coefficient matrix for the original system, and
the coefficient matrix of the variable transformation:

A = [ a11 a12 ]   B = [ b11 b12 ]
[ a21 a22 ] ,     [ b21 b22 ] .

This type of combination of two matrices is an important operation on
matrices, and we call it "matrix multiplication", because we find that
it has many properties of what we would expect from a multiplication
(although it is not commutative).

If you have any questions, please write back and I will try to explain
further.

- Doctor Fenton, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Algebra

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search