16 Robert Beezer §TSS
M45 (Robert Beezer) The details for Archetype J include several sample solutions. Verify that one of
these solutions is correct (any one, but just one). Based only on this evidence, and especially without doing
any row operations, explain how you know this system of linear equations has infinitely many solutions.
Solution (Robert Beezer) Demonstrate that the system is consistent by verifying any one of the four
sample solutions provided. Then because n = 9 > 6 = m, Theorem CMVEI gives us the conclusion that the
system has infinitely many solutions. Notice that we only know the system will have at least 9 − 6 = 3 free
variables, but very well could have more. We do not know know that r = 6, only that r ≤ 6.
M46 (Manley Perkel) Consider Archetype J, and specifically the row-reduced version of the augmented
matrix of the system of equations, denoted as B here, and the values of r, D and F immediately following.
Determine the values of the entries
[B]
1,d
1
[B]
3,d
3
[B]
1,d
3
[B]
3,d
1
[B]
d
1
,1
[B]
d
3
,3
[B]
d
1
,3
[B]
d
3
,1
[B]
1,f
1
[B]
3,f
1
(See Exercise TSS.M70 for a generalization.)
For Exercises M51–M57 say as much as possible about each system’s solution set. Be sure to make it
clear which theorems you are using to reach your conclusions.
M51 (Robert Beezer) A consistent system of 8 equations in 6 variables.
Solution (Robert Beezer) Consistent means there is at least one solution (Definition CS). It will have
either a unique solution or infinitely many solutions (Theorem PSSLS).
M52 (Robert Beezer) A consistent system of 6 equations in 8 variables.
Solution (Robert Beezer) With 6 rows in the augmented matrix, the row-reduced version will have
r ≤ 6. Since the system is consistent, apply Theorem CSRN to see that n −r ≥ 2 implies infinitely many
solutions.
M53 (Robert Beezer) A system of 5 equations in 9 variables.
Solution (Robert Beezer) The system could be inconsistent. If it is consistent, then because it has
more variables than equations Theorem CMVEI implies that there would be infinitely many solutions.
So, of all the possibilities in Theorem PSSLS, only the case of a unique solution can be ruled out.
M54 (Robert Beezer) A system with 12 equations in 35 variables.
Solution (Robert Beezer) The system could be inconsistent. If it is consistent, then Theorem CMVEI
tells us the solution set will be infinite. So we can be certain that there is not a unique solution.
M56 (Robert Beezer) A system with 6 equations in 12 variables.
Solution (Robert Beezer) The system could be inconsistent. If it is consistent, and since 12 > 6, then
Theorem CMVEI says we will have infinitely many solutions. So there are two possibilities. Theorem
PSSLS allows to state equivalently that a unique solution is an impossibility.
M57 (Robert Beezer) A system with 8 equations and 6 variables. The reduced row-echelon form of the
augmented matrix of the system has 7 pivot columns.
Solution (Robert Beezer) 7 pivot columns implies that there are r = 7 nonzero rows (so row 8 is all
zeros in the reduced row-echelon form). Then n + 1 = 6 + 1 = 7 = r and Theorem ISRN allows to
conclude that the system is inconsistent.
M60 (Robert Beezer) Without doing any computations, and without examining any solutions, say as much
as possible about the form of the solution set for each archetype that is a system of equations.
Archetype A, Archetype B, Archetype C, Archetype D, Archetype E, Archetype F, Archetype G, Archetype
H, Archetype I, Archetype J