Algebra Know-It-ALL

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.