SOLUTION-MANUAL-FINITE-MATHEMATICS-AND-ITS-APPLICATION-9TH-EDITION-GOLDSTEIN

GOLDSTEIN
Chapter 2 n




Instructor’snSolutionnManual
Exercisesn2.1 ⎨
⎩n–5xn+n4yn=1
⎧⎪n 1nxn–n3yn=n2 [2]+5[1]nnn ⎧nxn+n2yn=n3
1. ⎨n2 ⎯n⎯⎯n ⎯⎯n →n⎨ ⎩
⎪n⎩n5x+n4yn=n1 14y
⎧nxn–n6yn=n4
2[1]
=16
⎯n⎯⎯n →n⎨
⎩n5xn+n4yn=1
4. –n n(first)n –n nxn+3yn=n–2

⎧nxn+n4yn=n6 (–1)[2]
⎧nxn+n4yn=n6 +(second) xn+n2yn=1
2. ⎨n ⎯n⎯⎯n ⎯n→n⎨ 5yn=n–1
⎩ –nyn=n2 ⎩ yn=n–2
⎧n⎪nxn–n6yn=n4 [2]+ ( )–21 [1]
n n ⎧ x – 6yn=n4
nnn n n n

3. 5(first)n 5x+10yn=15n+n(second) – ⎨n1nx+n2yn=1n⎯n⎯⎯n ⎯⎯n ⎯n→n⎨n⎩ 5yn=n–1
5x+n4yn=1
⎩n⎪n2

14yn=16 5. –4(first)n –4xn+8y–n4zn=n0n+n(third)n4xn+ny
⎧ xn+2yn=n3 +3zn=n5

36

,ISM:nFinitenMath Chaptern2:nMatrices


19. Usen[1]nton(–
2)[2]ntonchangenthen2ntonan0.
9yn–nzn=n5
⎧nxn–n2y+nzn=n0 20. Usen[3]n+n(–
⎪ 4)[1]ntonchangenthen4ntonan0.
⎨ yn–n2zn=n4
21. Interchangenrowsn1nandn2nornrows
⎪n 1nandn3ntonmakenthenfirstnentryninn
⎩n4xn+ny+n3zn=n5
rown1nnonzero.
⎧nxn–n2y+nzn=n0
⎪
[3]+(–4)[1] ⎛n1⎞
⎯n⎯⎯n ⎯⎯n ⎯n→n⎨ yn–n2zn=n4 22. Usen⎜n– ⎟n[2]ntonchangenthen–
3ntonan1.
⎪n
⎩ 9y–nzn=n5 ⎝n3⎠

6. 3(second)n 3y+n9zn=n3 23. Usen[1]n+n(–
+(third)n –3y+n7zn=n2 3)[3]ntonchangenthen3ntonan0.

24. Interchangenrowsn2nandn3ntonmaken
16zn=n5 thensecondnentryninnrown2nnonzer
⎧nxn+6yn–n4zn=1 ⎧nxn+n6yn–n4zn=1 o.
⎪ [3]+3[2] ⎪
⎡n3 96⎤n 1
⎨ yn+3zn=1n⎯n⎯⎯n ⎯⎯n →n⎨ yn+3zn=1
25.nnn ⎢n⎣n2 86⎥n⎦n⎯n⎯⎯
⎪n ⎪n
⎩n –3y+n7zn=n2 ⎩ 16zn=n5 3[1]→n⎡n⎢n⎣n21 38 26⎤n⎥n⎦
⎡n1n–n1
7. ⎢n⎣n0 21 43⎤n⎦n⎥n⎯n⎯⎯ [2]+(–2)[1]⎡n1nnn 3 2⎤
[1]+12⎯⎯n [2]→n⎣n⎡n⎢ 01 ⎯n ⎯⎯n ⎯⎯n ⎯n →n ⎢n ⎣n 0n2n 2⎥n ⎦
0154⎥⎤n⎦ 7
1
⎡ 1 3 2⎤
2[2]n n n n
–2
8 ⎯n⎯⎯n →n⎢n⎣n0n 1n 1⎥n⎦
⎡n1 09⎤
⎥ [1]+(–3)[2] ⎡n1nnn 0 –1⎤
⎢
8. 0 13 ⎯n⎯⎯n ⎯⎯n ⎯n →
⎢n⎥
⎢n⎣n0 45⎦n⎥ ⎢n⎣n0 1
⎡n1nnn 0 9⎤
[3]+(–4)[2] ⎢ 1 ⎥ 1⎥n⎦n xn=n–
⎯n⎯⎯n ⎯⎯n ⎯n→n0n 0 3
7n– ⎥
⎢ 1,nyn=n1
2n16n
⎢n⎣n0 –7⎥
⎦
17. Usen[2]n+n2[1]ntonchangenthen–
2ntonan0.


18. .nUsen n[2]ntonchangenthen2ntonan1.




37

,ISM:nFinitenMath Chaptern2:nMatrices




26.nnn ⎢ ⎥
⎡n13 261⎤n⎦n⎯n⎯⎯ 3[1]→n⎡n⎢n–21 –4663⎤n⎥n⎦n 28.n⎢n⎢ ⎡n–4 –71
03–314⎥n⎤n⎥
⎣n–2 –4⎣
[2]+2[1] ⎡n1nnn6 3⎤ ⎢n⎣n6nnn 14 750⎥n⎦

⎯n⎯⎯n ⎯⎯n →n⎢n⎣n08n 12⎥n⎦ ⎡n1 2 08⎤

1[2] ⎡n1nnn 6 3⎤ ⎯n⎯⎯n 2[1]→n⎢n–4–7 3–31⎥
8⎢n⎥
0
⎯n⎯⎯n →n⎢n⎣n0n 132n⎥n⎦ ⎢n⎣n6nnn 14 3 750⎥n⎦
1 0–6
[1]+(–6)[2] ⎡n ⎤ ⎡n1n 28⎤nnn 7

⎯n⎯⎯n ⎯⎯n ⎯ →n⎢n⎣n0 132⎥n⎦nnn ⎯n⎯⎯n [2]+ 4[1]⎯⎯

→n⎢n⎢n0 11⎥n⎥

xn=n–6,nyn= ⎢n⎣n6n1450⎥n⎦
⎡n1n2n08
⎤
[3]+(–6)[1] ⎢n⎥
⎡n1 –3 41⎤ ⎯n⎯⎯n ⎯⎯n ⎯n→n⎢n0n 1
31⎥

⎢n ⎥
27.nnn 4n –10 104 ⎢n⎣n0 2 72⎥n⎦
⎢n⎥
⎢n⎣n–3 9 −5–6⎥n⎦ [1]+(–2)[2]nnn ⎢n⎡n1n 0 –66⎤n⎥

⎡n1 –3 4 1⎤ ⎯n⎯⎯ ⎯⎯ ⎯n→n⎢n0 1 31⎥
[2]+(–4)[1] ⎢n⎥ ⎢n⎣n0 2 72⎥n⎦
⎯n⎯⎯n ⎯⎯n ⎯n→n 0 2 –6 0
⎢n⎥
⎢n⎣n–3 9 –5 –6⎥n⎦ [3]+(–2)[2]n ⎢n⎡n1n 0 –66⎤n⎥

⎡n1 –3 4 1⎤ ⎯n⎯⎯n ⎯⎯ ⎯n→n⎢n0 1 31⎥
[3]+3[1] ⎢n⎥ ⎢n⎣n0 0 10⎥n⎦
⎯n⎯⎯n ⎯→n 0 2 –6 0
⎢n⎥
⎢n⎣n0 0 7 –3⎥n⎦ [1]+6[3] ⎢n⎡n1n 0 0 6⎤n⎥

⎡n1n –3 4 1⎤ ⎯n⎯⎯n ⎯→n⎢n0 1 3 1⎥

⎯n⎯⎯n 12[2]→n⎢n0n 1 –3 0⎥ ⎢n⎣n0 0 1 0⎥n⎦




38

, ISM:nFinitenMath Chaptern2:nMatrices




⎢ ⎥
⎢n⎣n0 0 7n –3⎥n⎦ [2]+(–3)[3]nnn ⎢n⎡n1n 0 0n 6⎤n⎥

⎡n1nnn 0 –5 1⎤ ⎯n⎯⎯n ⎯⎯n ⎯n→n⎢n0n 1nnn 0 1⎥
[1]+3[2]⎢ ⎥ ⎢n⎣n0n 0 1n 0⎥n⎦
⎯n⎯⎯n ⎯→n 0 1nnn –3 0
⎢n ⎥n
⎢n⎣n0n 0 7n –3 ⎥n⎦ xn=n6,nyn=n1,nzn=n0


⎯n⎯⎯n 1[3]→n⎡n⎢n⎢n01n 01 –3–5 01⎤n⎥n⎥ 29.n⎣n⎢n⎡n23n –24–
41⎥n⎦n⎤n⎯n⎯⎯n 21[1]→n⎢n⎣n⎡n31–14–21⎥⎤n⎦
7



⎢n⎣n0n 0 1nnn – ⎥n⎦ [2]+(–3)[1] ⎡n1nnn –1nnn –2⎤
⎯n⎯⎯n ⎯⎯n ⎯n→n⎢
⎡n1nnn 0 0 –n78⎤ ⎣n0 7 7⎥n⎦

[1]+5[3] ⎢n⎥ 1[2] ⎡n1nnn –1nnn –2⎤
3
⎯n⎯⎯n ⎯→n ⎢n⎢n⎣n00n 01n 7 –31 3
0⎥n⎥n⎦n ⎯n⎯⎯nn7→n⎢n⎣n0 1
1⎥n⎦
–n7


[2]+3[3]nnn ⎡n⎢n1n 0nnn 0n –n879n⎤n⎥ ⎯n⎯⎯n [1]+1[2]⎯→n ⎣n⎢n⎡n01 01
–11⎤n⎥⎦
3
7

⎯n⎯⎯n ⎯⎯n →n⎢n0n 1nnn 0 –n7n⎥ xn=n–1,nyn=n1
⎢n⎣n0n 0 1 –n ⎥n⎦

xn=n– ,nyn=n– ,nzn=n–

1 2
30.n⎢n⎣n⎡n–12n 23n –24⎥n⎦n⎤n⎯n⎯⎯ 21[1]
→n⎡n⎢n⎣n–1 2n –2 ⎦n⎤n⎥n 32.n⎢n⎢ ⎡n11n–12 –12
–411⎥n⎥n⎤

⎡n ⎤
[2]+1[1]n n 1 2 ⎢n⎣n2 5 939⎥n⎦




39