Download MATH 107.01 HOMEWORK #1 SOLUTIONS Problem 1.1.2. Solve

MATH 107.01
HOMEWORK #1 SOLUTIONS
Problem 1.1.2. Solve the system
2x + y − 2z = 0
2x − y − 2z = 0
x + 2y − 4z = 0
Solution. Note that

2
rref 2
1
 
1 −2 0
1
−1 −2 0 = 0
2 −4 0
0
0
1
0
0
0
1

0
0
0
Hence the only solution is
   
0
x
y  = 0 .
0
z
Problem 1.1.8. Solve the system
x1 + x2 − x3 + 2x4 = 1
x1 + x2 − x3 − x4 = −1
x1 + 2x2 + x3 + 2x4 = −1
2x1 + 2x2 + x3 + x4 = 2
Solution. Note that

1 1
1 1
rref 
1 2
2 2
 
1 0 0 0
−1
2
1
0 1 0 0
−1 −1 −1
=
1
2 −1 0 0 1 0
1
1
2
0 0 0 1

11/3
−10/3

2/3
2/3
Hence the only solution is
  

x1
11/3
x2  −10/3
 =

x3   2/3  .
x4
2/3
Problem 1.1.15. Solve the system
2x1 − x2 − x3 + x4 + x5 = 0
x1 − x2 + x3 + 2x4 − 3x5 = 0
3x1 − 2x2 − x3 − x4 + 2x5 = 0
Solution. Note that

2 −1
rref 1 −1
3 −2
 
1 0 0 7
−1
1
1 0
1
2 −3 0 = 0 1 0 9
0 0 1 4
−1 −1
2 0
1

−4 0
−5 0
−4 0
2
MATH 107.01 HOMEWORK #1 SOLUTIONS
Hence the solutions are given by

  
4x5 − 7x4
x1
x2  5x5 − 9x4 

  
x3  = 4x5 − 4x4  .

  

x4  
x4
x5
x5
Problem 1.1.17. Determine conditions on a, b, and c so that the system
2x − y + 3z = a
x − 3y + 2z = b
x + 2y + z = c
has a solution.
Solution. The row reduction


2 −1 3 a
1 −3 2 b
1
2 1 c
WV
PQ
2R2 → R2
2R3 → R3
R2 − R1 → R2
R3 − R1 → R3
WV
PQ

2
/ 2
2

2
/ 0
0

2
/ 0
0
R3 +R2 →R3
−1 3
−6 4
4 2

a
2b
RS
2c UT

−1
3
a
−5
1 2b − a
RS
5 −1 2c − a UT

−1 3
a

−5 1
2b − a
0 0 −2a + 2b + 2c
shows that the rref of the system is consistent if and only if −2a + 2b + 2c = 0.
Hence the system has a solution if and only if a − b − c = 0.
Problem 1.1.18. Determine conditions on a, b, and c so that the system
x + 2y − z = a
x + y − 2z = b
2x + y − 3z = c
has a solution.
Solution. The row reduction



R1 → R2
1
1 2 −1 a RR2 −−2R
3
1 → R3
/ 0
1 1 −1 b
2 1 −3 c
0
WV
PQ
R3 −3R2 →R3

1
/ 0
0
2 −1
−1 −2
−3 −1

a
b−a
RS
c − 2a UT

2 −1
a
−1 −2
b−a 
0
5 a − 3b + c
MATH 107.01
HOMEWORK #1 SOLUTIONS
3
shows that the rref of the system is consistent. Hence the system has a solution for
all values of a, b, and c.
Problem 1.1.19. Determine conditions on a, b, c, and d so that the system
x1 + x2 + x3 − x4 = a
x1 − x2 − x3 + x4 = b
x1 + x2 + x3 + x4 = c
x1 − x2 + x3 + x4 = d
has a solution.
Solution. The row reduction

 R2 − R1 → R2
1
1
1 −1 a
R3 − R1 → R3
1 −1 −1
R4 − R1 → R4
1 b


1
1
1
1 c
1 −1
1
1 d
WV
PQ
R4 −R2 →R4
WV
PQ
R3 ↔R4

1
0
/
0
0

1
1 −1
a
−2 −2
2 b − a

0
0
2 c − a
−2
0
2 d − a ml
RS

1
0
/
0
0

1
1 −1
a
−2 −2
2
b−a 

0
0
2
c−a 
0
2
0 d − 2a − b ml
RS

1
0
/
0
0

1
1 −1
a
−2 −2
2
b−a 

0
2
0 d − 2a − b
0
0
2
c−a
shows that the rref of the system is consistent. Hence the system has a solution for
all values of a, b, c, and d.
Problem 1.1.22. Determine if the homogeneous system
√
99x1 + πx2 − 5x3 = 0
2x1 + sin (1)x2 + 2x4 = 0
3.38x1 − ex3 + ln (2)x4 = 0
has a nontrivial solution.
Solution. This is a homogeneous system in 3 linear equations in 4 variables. Since
3 < 4, Theorem 1.1 in the book implies that this system has infinitely many nontrivial solutions.
Problem 1.1.23. Determine if the homogeneous system
x−y+z =0
2x + y + 2z = 0
3x − 5y + 3z = 0
has a nontrivial solution.
4
MATH 107.01 HOMEWORK #1 SOLUTIONS
Solution. Note that

1
rref 0
0
 
−1 1 0
1
1 2 0 = 0
−5 3 0
0
0
1
0
1
0
0

0
0 .
0
That is, the system is consistent and has free variable x3 . Hence the system has
infinitely many nontrivial solutions.
Problem 1.1.26. Give an example of a system of linear equations with more equations than variables that illustrates each of the following possibilities:
(a) Has exactly one solution.
(b) Has infinitely many solutions.
(c) Has no solution.
Solution. (a) The system
x=0
2x = 0
has more equations than variables and has x = 0 as a unique solution.
(b) The system
x+y =0
2x + 2y = 0
3x + 3y = 0
has more equations than variables. Furthermore,
 

1 1
1 1 0
rref 2 2 0 = 0 0
0 0
3 3 0

0
0 .
0
That is, the system is consistent and has free variable y. Hence the system has
infinitely many solutions.
(c) The system
x=0
x=2
has more equations than variables. Moreover, it has no solutions since 0 6= 2.