Algebra Know-It-ALL

Part Two 341

Answer 19-2

We can play the matrix morphing game to get the matrix into echelon form, which looks
like this:

####

0###

00##

where a pound sign (#) can represent any real number. Then we continue the game and go for
the diagonal form, which looks like this:

@00#

0@0 #

0 0@#

where a pound sign can represent any real number, and an at sign (@) can represent any
nonzero real number. Then we must get the matrix into the unit diagonal form, which looks
like this:

100 x

010 y

001 z

Once we have the matrix in the unit diagonal form, assuming we haven’t made any mistakes,
the values x,y, and z represent the solution to the original three-by-three linear system. We
can reduce the values we have found, if they are fractions, to their lowest forms. Finally, we
should plug our answers into the original equations to be sure we’ve arrived at the correct
solution to the system.

Question 19-3

How is the matrix morphing game played?

Answer 19-3

There are three types of moves:

• Swap two rows, while keeping the elements of both rows in the same order from left to
  right.


• Multiply or divide all the elements in any row by a nonzero constant, keeping the ele-
  ments in the same order from left to right.


• Add all the elements in any row to all the elements in another row, and then replace
  the elements in either row by the sum, keeping the elements of both rows in the same
  order from left to right.