Algebra Know-It-ALL

662 Worked-Out Solutions to Exercises: Chapters 11 to 19

effect, says that 0 is equal to some nonzero real number. That’s absurd! The reason for this

“hangup” is that the original three-by-three linear system is inconsistent. If we could draw

a graph of this system in Cartesian three-space, we’d get three parallel planes, no two of

which would intersect anywhere.

10. Here’s the three-by-three linear system we’ve been told to describe as a matrix:

x+y+z= 1

2 x+ 2 y+ 2 z= 2

3 x+ 3 y+ 3 z= 3

These equations are all in ideal form for conversion to

1111

2222

3333

Dividing the second row by 2 and the third row by 3, we get

1111

1111

1111

We won’t be able to make any of these elements vanish without making a whole row van-

ish, giving us the equation

0 x+ 0 y+ 0 z= 0

which is utterly useless. However, the above matrix tells us that

x+y+z= 1

An infinite number of ordered triples (x,y,z) satisfy this equation. Our original three-by-

three linear system is actually one equation stated three different ways. It’s redundant, so

a single solution does not exist.