The matrix form
Once we have morphed all the equations into the proper form, we can remove the variables
and equals signs. Then we can put the numbers into an array having three rows, each with
four numbers:
a 1 b 1 c 1 d 1
a 2 b 2 c 2 d 2
a 3 b 3 c 3 d 3Some texts enclose matrices in huge parentheses or brackets. But they aren’t really necessary, and
they can clutter things up. Let’s not use them.
Are you confused?
You might ask, “Can matrices represent two-by-two systems? Can they represent systems larger than three-
by-three? Can they represent asymmetrical systems such as five-by-four?” The answers are “Yes,” “Yes,”
and “Yes.” In this chapter, we’ll look at three-by-three systems only, but matrix techniques can be applied
to any linear system.
Here’s a challenge!
Put the following three-by-three linear system into matrix form:
7 y= 3 z+ 3
8 z=− 2 x− 7
12 x= 7 y
Solution
None of these three equations is in the proper form for conversion to matrix notation. We’ll have to
manipulate them. Here are the processes, step-by-step. For the first equation:
7 y= 3 z+ 3
0 x+ 7 y= 3 z+ 3
0 x+ 7 y− 3 z= 3
For the second equation:
8 z=− 2 x− 7
2 x+ 8 z=− 7
2 x+ 0 y+ 8 z=− 7
For the third equation:
12 x= 7 y
12 x− 7 y= 0
12 x− 7 y+ 0 z= 0
How to Build a Matrix 297