GAUSS/JORDAN EXAMPLE "c"
FEWER VARIABLES THAN EQUATIONS
When there are fewer variables than equations, we proceed just as we did in
other examples of GAUSS/JORDAN. The GAUSS/JORDAN
procedure will end, typically, in a matrix whose left part contains one or
more rows of zeros. The student's dilemma, therefore, is how to interpret
such rows of zeros. An example is below, with 3 equations and
2 variables; watch what happens, especially near the end :
Below is the system of equations which we will
solve by G/J 


Below is the 1st augmented matrix:
pivot
on "1" in the 11 position 

row operations P2 for the
first pivoting are named below 
Next we pivot on the
number "7" in the 22 position, encircled in red below 


Below is the result of performing
P1 on the element "7" in
the 22 position. Next we must perform
P2 

The row operations of P2
are below 
The result of the second and last pivoting is below. The
remaining row contains only zeros, so the row operations of
GAUSS/JORDAN are complete 


Below is a copy of the final matrix of the previous line,
with it's identity matrix in blue 

We now rewrite each row of this matrix as an equation, just
as we wrote the original equations as a matrix 
The 3rd row below corresponds to an identity, giving no
information. Thus, we ignore row 3, basing a
final solution on rows 1 and 2 


IMPORTANT NOTE :
If the 3rd row of the above solution had been "0x+0y=2",
instead of "0x+0y=0", then the original problem would have
NO SOLUTION, since "0=2" is always false.
See also our text (Rolf): example 5 on page 113. 

Recall our original
system colorcoded 
x + 2y = 3
2x  3y =  8
x  4y =  9 


The 3 original equations are graphed below, showing
the solution found by G / J above.

