Gaussian elimination is a systematic strategy for solving a set of
linear equations. It can also be used to construct the inverse of a
matrix and to factor a matrix into the product of lower and upper
triangular matrices. We start by solving the linear system
Basically, the objective of Gaussian elimination is to do
transformations on the equations that do not change the solution, but
systematically zero out (eliminate) the offdiagonal coefficients,
leaving a set of equations from which we can read off the answers.
We express the problem in terms of a set of equations, and
sidebyside, we express it in terms of an equivalent matrix product.
We do this to show how the manipulations in the matrix tracks the
manipulations of the equations, where it is easier to see that we are
not changing the solution.
The method has two parts. First ``triangulation'' and then ``back
substitution''.
Equations 
Matrixvector representation 

Equivalent matrixvector equation


First step: examine the coefficients of . Swap equations
(1) and (3) so the largest coefficient is in the first equation
(first row). This is called the ``pivot'' element.


Swap the first the third rows of the matrix and
the first and third elements of the vector on the
right side.
Note that in the matrix equation we don't
interchange and .


Next step: Divide the first equation by the coefficient to
make the pivot element equal to .



Next step: Multiply the first equation by 2 and subtract it from the
second equation, putting the result in the second equation. This
``eliminates'' the coefficient of in the second equation.



Next step: Eliminate the coefficient of in the third equation
by subtracting the first equation from the third, putting the result
into the third equation.



Next step: Now work on the second column (coefficients ). We
want the largest coefficient in the second equation (diagonal element
in the matrix.) So swap the second and third equation.



Now divide by 5/3, the pivot element in the second column.



Now eliminate the coefficient of in the third equation by
multiplying the second equation by 1/3, subtracting the result from
the third equation, and putting the result in the third equation.



To complete the ``triangulation'' step we divide the third equation
by the coefficient of , namely .


Notice that the matrix is now in upper triangular form 
all elements below the diagonal are zero.



Notice that the matrix is now in upper triangular form 
all elements below the diagonal are zero.


Continuing on the third column, eliminate the coefficient of
in equation 1.



Next, work on the second column. We have only the coefficient of
in the first equation to eliminate. We then get the answer.


Notice that we now have a unit matrix, so
the solution can be read off.


The last step, of course, is to check the solution by plugging it in
to the orginal system of equations.


