Matrix Division?

Date: 06/26/2006 at 14:19:33
From: Randy
Subject: Dividing Matrices

Is it possible to divide matrices?  For example, would

  [3 4]
  [1 3]     [-1/2    -1/14]                                
  -----  =  [-7/12  -13/56]
  [6 7]        
  [4 8]

because I did the opposite of multiplication and divided both the
first and last numbers for the top row in the numerator and the column
in the denominator to get:

  3/6 - 4/4 =  -1/2
  3/7 - 4/8 =  -1/14
  1/6 - 3/4 =  -7/12
  1/7 - 3/8 = -13/56

Is that correct?

Date: 06/27/2006 at 12:38:44
From: Doctor Ricky
Subject: Re: Dividing Matrices

Hey Randy,

Actually, matrix division is not possible.  We can add, subtract, 
multiply and exponentiate matrices, but we cannot divide two matrices.  
This is due to an idea called matrix singularity, which arises from a 
branch of mathematics called linear algebra.  

If a matrix is singular, that means that it has only one possible 
solution (something that we need in order to define something 
explicitly) which is true in matrix addition, subtraction, 
multiplication and exponentiation, but not division.

An example of this is basic.  Let

  A = [ 1 0 ]
      [ 1 0 ]

and let A*B equal the zero matrix for some matrix B.

  i.e.  [ 1 0 ] * B   =  [ 0 0 ]
        [ 1 0 ]          [ 0 0 ]

If matrix division existed, that would mean that there would be only 
one matrix B that would make that true (i.e. B would be singular).  
In other words,

  B  =  [ 0 0 ]
        [ 0 0 ]
        -------         for lack of better notation
        [ 1 0 ]
        [ 1 0 ]

However, we can come up with a couple different examples that work 
for B.

   i.e.  B =  [ 0 0 ]   or  B = [ 0 0 ]   both work (you can check).
              [ 0 0 ]           [ 1 1 ]
  
Therefore, since there is not just one possible answer, it wouldn't 
make sense or be possible to define matrix division.

Hopefully this answered your question, but if you have any more 
questions, please let me know!

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