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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 
involves multiplying and adding to get each entry?  Adding, 
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


  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 

   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 

- Doctor Fenton, The Math Forum 
Associated Topics:
High School Linear Algebra

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