Part Two 345
Question 19-8
Suppose that we play the matrix morphing game in an attempt to solve a three-by-three linear
system, and we come up with a matrix that looks like this:
000 − 5
1112
0007
At first, we suspect we made a mistake somewhere. But when we try the game again, we get
the same matrix. What is the reason for this absurd result?
Answer 19-8
The original system of equations is inconsistent. There is no unique solution.
Question 19-9
Suppose that we play the matrix morphing game in an attempt to solve another three-by-three
linear system, and we come up with this elegant but useless matrix:
1111
1111
1111
What does this tell us?
Answer 19-9
The original system of equations is redundant. There are infinitely many solutions.
Question 19-10
Can the matrix morphing game be used to solve larger systems, such as four-by-four, five-by-
five, and so on?
Answer 19-10
Yes. The matrix morphing game can be applied to linear systems of any finite dimension.
That’s the good news. There’s bad news, too: The number of steps in the process increases
much faster than the dimension. If carried out “manually” on a large system, matrix morphing
is no game. It’s more like slow torture! But there’s some more good news: Computers can be
programmed to play the matrix morphing game quickly and without pain, even for gigantic
linear systems.