Algebra Know-It-ALL

306 The Matrix Morphing Game

When you get the unit diagonal form, you should end up with sloppy fractions in the far-right column.

When you reduce these fractions, you should get

1002

010 − 1

0013

Solution

You’re on your own. Have fun!

Practice Exercises

This is an open-book quiz. You may (and should) refer to the text as you solve these problems.

Don’t hurry! You’ll find worked-out answers in App. B. The solutions in the appendix may not

represent the only way a problem can be figured out. If you think you can solve a particular

problem in a quicker or better way than you see there, by all means try it!

1. Put the following three-by-three linear system into the proper form for conversion to
   matrix notation:
   x=y−z− 7
   y= 2 x+ 2 z+ 2
   z= 3 x− 5 y+ 4


2. Write the set of equations from the solution to Prob. 1 in the form of a matrix.


3. Write the set of equations represented by the following matrix:

(^04) − 1 − 2
(^5) −3/2 81
1111

4. Put the matrix of Prob. 3 into echelon form.


5. Put the matrix derived in the solution to Prob. 4 into diagonal form.


6. Reduce the matrix derived in the solution to Prob. 5 to a form with the smallest possible
   absolute values in each row, such that all the numbers in the matrix are integers.


7. Reduce the matrix derived in the solution to Prob. 6 to unit diagonal form. Then state
   the tentative solution to the three-by-three linear system we derived from the matrix in
   Prob. 3 and stated in solution 3.


8. Check the values for x,y, and z derived in the solution to Prob. 7 to be sure they’re
   correct. To do this, plug the numbers into the equations stated in the solution to Prob. 3.