Test Bank For Elementary Linear Algebra - 8th - 2017 All Chapters - 9781305658004.pdf

SOLUTIONS TO PROBLEMS
h h




ELEMENTARY LINE h




ARALGEBRA
h




DEPARTMENThOFhMATHEMATICS




FLORIDAhSTATEhUNIVERSITY

, CONTENTS
PROBLEMSh 1.6 .............................................1

PROBLEMSh 2.4 ........................................... 12

PROBLEMSh 2.7 ........................................... 18

PROBLEMSh 3.6 ........................................... 32

PROBLEMSh 4.1 ........................................... 45

PROBLEMSh 5.8 ........................................... 58

PROBLEMSh 6.3 ........................................... 69

PROBLEMSh 7.3 ........................................... 83

PROBLEMSh 8.8 ........................................... 91




i

, SECTION 1.6 h

Σ Σ Σ ΣΣ Σ
0hhhh0hhh 2hhhh4hhh 1 2 0
2.h(i) R1h↔hR2 R1h→h 2 R1
1h
;
0 0 0 0 0
Σ 2hhhh4hhh Σ 0hhhh0hhh Σ Σ
0 0
Σ Σ
0hhhh1hhh 1hhhh2hhh 4 1hhh 0hhh −2
(ii) 3 R1h↔hR2 R1h→hR1h−h2R ;
0hhhh1hhh3 0hhh 1 3
1hhhh2hhh 2

4
1h1h1 R2h→hR2h−hR1 1 1 0
(iii)h h1 1 0 h0 0h−1h
R3h→hR3h−hR1
1 0 0 0h−1h −1
R1h→hR1h+hR3 1hhh 0 0 R2h→hR2h+hR 1 0 0
R3h→h−R3 h0hhh 1 1h 3
h
h
0h1h0h ;
R3h→h−R3
R2h↔h R3 0h0h−1 0 0 1
2h0h0 1h0 0
R3h→hR3h+h2R1
(iv)h 0h0h0h h 0h0 0h .
R1h→hR12hhhh 1
− 4h0h0 0h 0 0
1 1 1 2 R2h→hR2h−h2R1 1 1 1 2
3.h(a)h h 2 3hhhh−1 8h 0 1 −3 4
R3h→hR3h−hR1
1h−1h−1h−8 0 −2 −2 −10

R1h→hR1h−hR2 1hhh 0 4hhhh−2 1hhh 0 4 2
R3h→hh−1hR3h
h
0hhh 1hhh −3
hh 4hh h 0hh 1hhh −3 4h
hh

R3h→hR3h+h2R2 8
0h0h−8h−2 0hhh 0 1 1
4

1h0h0h−3
R1h→hR1h−h4R3
0hh 1hhh 0
hh 19
.
R2h→hR2h+h3R3 4
0hhh 0hhh 1 1
4
Thehaugmentedhmatrixhhashbeenhconvertedhtohreducedhrow–
echelonhformhandhwehreadhoffhthehuniquehsolutionh4xh=h−3,h4yh=h19h,hzh=h1h.
1 1 −1 2h 10 R2h→hR2h−h3R1 1 1 −1 2 10
(b) 3hhh −1 7 4 1h 0hhh −4
h 10h−2h−29h
R3h→hR3 +h5R1
−5 3hhh −15hhh −6 9 0 8hhh −20 4 59
1 1hhhh−1 2 10
R3h→hR3h+h2R2h h
0hhh −4 10h−2h−29h .
0 0 0 0 1

1

, Fromhthehlasthmatrixhwehseehthaththehoriginalhsystemhishinconsistent.




2