Algebra Know-It-ALL

300 The Matrix Morphing Game

The final goal

The matrix morphing game, like any sensible game, has an ultimate objective. Our goal is to

get a matrix representing a three-by-three linear system into this form:

100 x

010 y

001 z

where x,y, and z are real numbers. This is called the unit diagonal form.

Now imagine that we start with a three-by-three linear system, make it into a matrix, and

then play the matrix morphing game until we get the unit diagonal form. Do you suspect that

the values x,y, and z, which appear in the far right column, will represent the solution to the

linear system? If so, you are right, provided the system is consistent (has a unique solution).

Are you confused?

Do you wonder why we can’t multiply all the elements in a row by 0? Think about this for a minute. Doing

that would wipe out one of the equations in the system, leaving us with a two-equation system having

three variables. Such a system doesn’t contain enough information to define a unique solution.

Are you still confused?

You might also ask, “Why can’t we add two rows in a matrix and then replace the remaining row (the one

not involved in the sum) with that sum?” Well, we could try it, and for awhile it might seem to work. But

such a move would delete the information in the equation represented by the replaced row, turning it into a

mere hybrid of the other two rows. We would be left with the equivalent of a two-equation system in three

variables. We might end up thinking that the original system was redundant, when in fact it was consistent.

A Sample Problem

In this section, we’ll solve a three-by-three linear system using matrix operations. The game

can be played in many ways. The process in this section doesn’t represent the only avenue by

which the final result can be reached. But if we manage to avoid making mistakes, we’ll always

get to the same destination, no matter what road we take!

The game plan

Anyone who plays a game needs a plan. Here’s a strategy for the matrix morphing game that

works well for me. First, we get equations into form:

a 1 x+b 1 y+c 1 z=d 1

a 2 x+b 2 y+c 2 z=d 2

a 3 x+b 3 y+c 3 z=d 3