Algebra Know-It-ALL

These equations are all in ideal form for conversion to the matrix

1111

1112

1113

Now let’s try to get this into unit diagonal form. The first step along the way is to seek the

echelon form. We can start by doing one of three things:

• Make the first entry in the second row vanish


• Make the first entry in the third row vanish


• Make the second entry in the third row vanish

This is easy—too easy! Suppose we want to make the first entry in the third row vanish.

We can multiply the first row through by −1, getting

− 1 − 1 − 1 − 1

1112

1113

Adding the first row to the third row and then replacing the third row with the sum gives us

− 1 − 1 − 1 − 1

1112

0002

That takes care of not only one, but two of the elements we wanted to turn into 0. But

there’s a problem starting to take shape. We want the third entry in the third row to end

up as a nonzero element. We won’t be able to do that without making both the first and

the second elements in that row nonzero as well. We can go further and add the first two

rows in the above matrix together, replacing the first row with the sum. Then we get

0001

1112

0002

This in effect states the following three equations:

0 x+ 0 y+ 0 z= 1

x+y+z= 2

0 x+ 0 y+ 0 z= 2

There are no real numbers x,y, or z such that, when they are each multiplied by 0, the

result is 1 or 2. No matter how we approach this problem, we’ll get a statement that, in

Chapter 19 661