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.
- 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.