Algebra Know-It-ALL

Then we make the matrix:

a 1 b 1 c 1 d 1

a 2 b 2 c 2 d 2

a 3 b 3 c 3 d 3

Next, we get the matrix into echelon form, which looks like this:

####

0###

00##

where a pound sign (#) can represent any real number. Then we go for the diagonal form:

@00#

0@0 #

0 0@#

where a pound sign can represent any real number, and an at sign (@) can represent any non-
zero real number. Our final goal is the unit diagonal form:

100 x

010 y

001 z

Once we’ve put a matrix into the unit diagonal form, we may find that one or more of the
solutionsx,y, or z is a fraction that can be reduced. We ought to reduce all solutions to their
lowest forms in the interest of elegance.

What if we can’t “win”?

If we find it impossible to get a matrix into the unit diagonal form, our failure indicates one
of four things:

• We didn’t try hard enough


• We made a mistake somewhere


• The original system is inconsistent


• The original system is redundant

The system to be solved

Now let’s methodically tackle a three-by-three linear system and solve it using the matrix mor-
phing game. Here are the equations:

3 x+z= 2 y+ 11

4 y+ 2 z=x

− 5 x+y= 3 z− 20

A Sample Problem 301