C()
1
1.1A
4. (c) 17 .
5. (a) Jim .
Joe .
(b) Joe .
Jim Joe .
2
2.1A
1. (a) −x1 + x2 1.
(c) x1 − x2 0.
(e) 0.5x1 − 0.5x2 0.
3. M1 = 4 /.
2.2A
1. (a e) C.1.
2. (a d) C.2.
5. x1 = A x2 = B max z = 20x1 + 50x2
s.t.
−0.2x1 + 0.8x2 0,
x1 100,
x1 , x2 0
(x1 , x2 ) = (80, 20), z = 2 600 .
, * .
2x1 + 4x2 240
2
C() 7. C.1
C.2
x1 = A x2 = B max z = 0.05x1 + 0.08x2
s.t. 0.75x1 − 0.25x2 0,
x1 − 0.5x2 0,
0.5x1 − 0.5x2 0
x1 + x2 5 000
x1 , x2 0
(x1 , x2 ) = (2 500, 2 500), z = 325 .
11. x1 = x2 = max z = 2x1 + x2
s.t. x1 + x2 10, x1 − x2 0
x1 4, x1 , x2 0
(x1 , x2 ) = (4, 6), z = 14.
14. x1 = C1 x2 = C2 max z = 12 000x1 + 9 000x2
s.t. −200x1 + 100x2 0, 2.1x1 + 0.9x2 20
x1 , x2 0
(x1 , x2 ) = (5.13, 10.26), z = 153 846 .
(a) C1:C2=0.5.
C() (b) , 7 692 /.
18. x1 = HiFi1 x2 = HiFi2 min z = 1 267.2 − (15x1 + 15x2 )
s.t.
6x1 + 4x2 432,
4x1 + 6x2 422.4,
5x1 + 5x2 412.8
x1 , x2 0
(x1 , x2 ) = (50.88, 31.68), z = 28.8 .
2.2B
1. (a) C.3.
C.3
5. x1 = /
x2 = /
min z = x1 + x2
s.t.
−0.6x1 + 0.4x2 0,
0.2x1 + 0.1x2 14
0.25x1 + 0.6x2 30,
0.1x1 + 0.15x2 10
0.15x1 + 0.1x2 8,
x1 = 55, x2 = 30, z = 85.
7. x1 , x2 0
3
4
C() x1 = A x2 = B min z = 100x1 + 80x2
s.t.
0.03 0.06x1 + 0.03x2 0.06,
0.03 0.03x1 + 0.06x2 0.05
0.03 0.04x1 + 0.03x2 0.07,
x1 + x2 = 1,
x1 , x2 0
x1 = 0.33, x2 = 0.67, z = 86 667 .
2.3A
3. xij = i j max z = 0.05(4x11 + 3x12 + 2x13 ) + 0.07(3x22 + 2x23 + x24 )
+0.15(4x31 + 3x32 + 2x33 + x34 ) + 0.02(2x43 + x44 )
s.t. x11 + x12 + x13 = 1, x43 + x44 = 1
0.25 x22 + x23 + x24 + x25 1
0.25 x31 + x32 + x33 + x34 + x35 1
5x11 + 15x31 3, 5x12 + 8x22 + 15x32 6
5x13 + 8x23 + 15x33 + 1.2x43 7
8x24 + 15x34 + 1.2x44 7, 8x25 + 15x35 7
xij 0
x11 = 0.6, x12 = 0.4, x24 = 0.255, x25 = 0.025, x32 = 0.267, x33 = 0.387,
x34 = 0.346, x43 = 1, z = 523 750 .
2.3B
2. , p q. xij 2.3-2 , rij i j . , p = q ; p = q, , max
z=y
s.t.
xij ci , ∀i = j
p : I − y +
i = p :
j=p
rji xji =
j=i
xij 0
ampl2.3b-2.txt.
rjp xjp =
—— j=i
j=p
xij
xpj
C() 5
max z = y
s.t. :
xij ci , ∀ i = j
p : I +
rjp xjp =
xpj
j=p
q : y +
j=p
xqj =
j=q
i = p q:
rjq xjq
j=q
rji xji =
j=i
xij
j=i
xij 0
$ →$ 1.806 4%, $ → =C 1.796 6%, $ →£ 1.828 7%, $ → 2.851 5%, $ → KD 1.047 1%. , . , .
[ AMPL( ampl2.3b-2.txt) Excel ( solver
2.3b-2.xls) . 2.4 .]
2.3C
2. xi = i , i = 1, 2, 3, 4
yj = j , j = 1, 2, 3, 4, 5
max z = y5
s.t.
x1 + x2 + x4 + y1 10 000
0.5x1 + 0.6x2 − x3 + 0.4x4 + 1.065y1 − y2 = 0
0.3x1 + 0.2x2 + 0.8x3 + 0.6x4 + 1.065y2 − y3 = 0
1.8x1 + 1.5x2 + 1.9x3 + 1.8x4 + 1.065y3 − y4 = 0
1.2x1 + 1.3x2 + 0.8x3 + 0.95x4 + 1.065y4 − y5 = 0
x1 , x2 , x3 , x4 , y1 , y2 , y3 , y4 , y5 0
x1 = 0, x2 = $10 000, x3 = $6 000, x4 = 0,
y1 = 0, y2 = 0, y3 = $6 800, y4 = $33 642,
5 z = $53 628.73.
5. xiA = i A , i = 1, 2, 3
xiB = i B , i = 1, 2, 3
max z = 3x2B + 1.7x3A
s.t. x1A + x1B 100( 1 )
−1.7x1A + x2A + x2B = 0( 2 )
6
C() −3x1B − 1.7x2A + x3A = 0( 3 )
xiA , xiB 0,
i = 1, 2, 3
1 A $100 000, 2 B 170 00 .
.
2.3D
3. xj j , j = 1, 2, 3.
max z = 30x1 + 20x2 + 50x3
s.t.
2x1 + 3x2 + 5x3 4 000
4x1 + 2x2 + 7x3 6 000
x1 + 0.5x2 + 0.33x3 1 500
2x1 − 3x2 = 0
5x2 − 2x3 = 0
x1 200, x2 200, x3 150
x1 , x2 , x3 0
x1 = 324.32, x2 = 216.22, x3 = 540.54, z = 41 081.08 .
7. xij = j i , i = 1, 2, j = 1, 2, 3
Iij = j i , i = 1, 2, j = 1, 2, 3
3
min z =
(c1j x1j + c2j x2j + 0.2I1j + 0.4I2j )
j=1
s.t.
0.6x11 800, 0.6x12 700, 0.6x13 550
0.8x21 1 000, 0.8x22 850, 0.8x23 700
x1j + I1,j−1 = x2j + I1j , x2j + I2,j−1 = dj + I2j , j = 1, 2, 3
I1,0 = I2,0 = 0, 0
dj = 500, 450, 600, j = 1, 2, 3
c1j = 10, 12, 11, j = 1, 2, 3
c2j = 15, 18, 16, j = 1, 2, 3
x11 = 1 333.33 , x13 = 216.67 , x21 = 1 250 , x23 = 300 ,
z = 39 720 .
2.3E
2. xs = /, xb = /, xn = /, xw =
/.
C() min z = 1.1xs + 1.5xb +
s.t.
70 80
xn +
20 30
7
xw
y = xs + xb + xn + xw
y 1, xs 0.1y, xb 0.25y, xn 0.15y, xw 0.1y
1
1
10 xb xn ,
50 xb xw
z = 1.12 , y = 1, xs = 0.5, xb = 0.25, xn = 0.15, xw = 0.1.
5. xA = A /, xB = B /, xr = /, xp = /, xj = /.
+
+
max z = 50(xr − s+
r ) + 70(xp − sp ) + 120(xj − sj )
−
+
−
+
+
−(10s−
r + 15sp + 20sj + 2sr + 3sp + 4sj ) − (30xA + 40xB )
s.t. xA 2 500, xB 3 000,
xr = 0.2xA + 0.25xB , xp = 0.1xA + 0.3xB , xj = 0.25xA + 0.1xB
−
+
+
−
+
xr + s−
r − sr = 500, xp + sp − sp = 700, xj + sj − sj = 400
0
z = $21 852.94, xA = 1 176.47/, xB = 1 058.82/, xr =
500/, xp = 435.29/, xj = 400/, s−
p = 264.71/.
2.3F
1. xi (yi ) = i min z = 2
6
8 / (12 /)
xi + 3.5
i=1
6
.
yi
i=1
s.t. x1 + x6 + y1 + y5 + y6 4, x1 + x2 + y1 + y2 + y6 8,
x2 + x3 + y1 + y2 + y3 10, x3 + x4 + y2 + y3 + y4 7,
x4 + x5 + y3 + y4 + y5 12, x5 + x6 + y4 + y5 + y6 4
x1 = 4, x2 = 4, x4 = 2, x5 = 4, y3 = 6, = 0, z = 49. = 20. 8 , = 26, z = 2 × 26 = 52,
,(8 +12 ) .
5. xi = i (i = 1
8:01, i = 9
4:01.)
min z = x1 + x2 + x3 + x4 + x6 + x7 + x8 + x9
s.t. x1 2, x1 + x2 2, x1 + x2 + x3 3
C() 8
x2 + x3 + x4 4, x3 + x4 4, x4 + x6 3
x6 + x7 3, x6 + x7 + x8 3, x7 + x8 + x9 3
x5 = 0, 2 8:01, 1 10:01, 3 11:01, 9 .
3 2:01.
2.3G
2
.
(b) (3,0,0), (1,1,0), (1,0,1) (0,2,0), 0, 3, 1 1
(c) 20 30 .
(d) 20 50 .
1. (a) 1 150L
.
6.
gi , yi ri i . .
.
max z = 3(500/3 600)g1 + 4(600/3 600)g2 + 5(400/3 600)g3
s.t.
(500/3 600)g1 + (600/3 600)g2 + (400/3 600)g3
(510/3 600)(2.2 × 60 − 3 × 10)
g1 + g2 + g3 + 3 × 10 2.2 × 60, g1 25, g2 25, g3 25
g1 = 25 , g2 = 43.6 , g3 = 33.4 , = 58.04 /
.
2.4A
2. (d) AppenCFile solver2.4a-2(d).xls.
2.4B
2. (c) AppenCFile ampl2.4b-2(c).txt.
(f) AppenCFile ampl2.4b-2(f).txt.
3.1A
1. M1 M2, 2 / 1 /.
C() 4. xij =
9
j i .
max z = 10(x11 + x12 ) + 15(x21 + x22 )
x11 + x21 − x12 − x22 + s1 = 5
s.t.
−x11 − x21 + x12 + x22 + s2 = 5
x11 + x21 + s3 = 200
x12 + x22 + s4 = 250
si , xij 0, ∀ i, j
3.1B
3. xj = j , j = 1, 2, 3.
+
max z = 2x1 + 5x2 + 3x3 − 15x+
4 − 10x5
+
2x1 + x2 + 2x3 + x−
4 − x4 = 80
s.t.
+
x1 + x2 + 2x3 + x−
5 − x5 = 65
+
−
+
x1 , x2 , x3 , x−
4 , x4 , x5 , x5 0
x2 = 65 , x−
4 = 15 , = 0, z = 325 .
3.2A
1. (c) x1 = 67 , x2 =
(e)
12
7 ,
z=
48
7 .
(x1 = 0, x2 = 3) (x1 = 6, x2 = 0) .
3. (x1 , x2 ) =
26
3
, − 34 , (x1 , x3 ) = (8, −2) ,
(x1 , x4 ) = (6, −4) ,
(x2 , x3 ) = (16, −26) ,
(x2 , x4 ) = (3, −13) ,
(x3 , x4 ) = (6, −16) .
3.3A
3. (a) (A, B)
, AB
.
(b) (i) . (ii) , C I . (iii) , 5. (a) x3 1 ,
D z = 3.
3.3B
3.
x1
x2
x3
x4
1.5
1
0
0.8
x7
x7
x8
x5
. A.
10
C() 6. (b) x2 , x5 x6 z
. x2
, x8
,
∆z = 5 × 4 = 20.
x5
, x1
,
∆z = 0, , x5 0. x6
, , x6 . ∆z = ∞, x6 .
9. s2 , z = 20.
3.4A
3. (a) min z = (8M − 4)x1 + (6M − 1)x2 − M s2 − M s3 = 10M .
(b) min z = (3M − 4)x1 + (M − 1)x2 = 3M .
6. x1
x2
x3
x4
z
−1
−12
0
0
−8
x3
1
1
1
0
4
x4
1
4
0
1
8
3.4B
1. , .
II 7. I , , I ; , I .
3.5A
1. (a) A → B → C → D.
(b) A 1, B 1, C C42 = 6,
3.5B
1. 1
0, 0, 10
3 , (0, 5, 0), 1, 4, 3 . D 1.
1
α3 , 5α2 + 4α3 , 10
3 α1 + 3 α3 , α1 + α2 + α3 = 1, 0 αi 1, i = 1, 2, 3.
3.5C
2. (a) x2
.
(b) . x2 , z 10 .
3.5D
1. 275 .
C() 11
3.6A
2. x1 = 1 x2 = 2 max z = 8x1 + 5x2
s.t.
2x1 + x2 400
x1 150, x2 200,
x1 , x2 0
(a) C.4x1 = 100, x2 = 200, z = 1 800 (b) (200, 500) 2 4 .
, (c)
(d) B.
(100, ∞) 0 .
1 . 2 = 200.
(100, 400) C.4
3.6B
c1
c2 2.
cc12 = 1.
3. (a) 0 (b)
.
3.6C
2. (a) , = 1 (
10 ),
0.83 /.
(b) 2 / ( 400 )= 240 .
12
C() = 110 . = 130 .
, 0, .
(d) D1 = 10 . D1 10, = 1 /. x1 = 0, x2 = 105,
x3 = 230, = ($1 350 + $1 × 10) − $40
60 × 10 = $1 353.33.
(e) D2 = −15. D2 −20, = 2 /. = 30 . = 7.5 . .
6. x1 = , x2 = , x3 = (c) max z = x1 + 50x2 + 5x3
s.t.
15x1 + 300x2 + 50x3 + s1 = 10 000, x3 − S2 = 5,
x1 + s3 = 400, −x1 + 2x2 + s4 = 0, x1 , x2 , x3 0,
s1 , S 2 , s3 , s4 0
(a) x1 = 59.09 , x2 = 29.55 , x3 = 5
, z = 1 561.36.
(b) TORA, z + 0.158s1 + 2.879S2 + 0s3 + 1.364s4 = 156.364. (c)
11. (a)
(b)
(c)
13. (b)
0.158, −2.879, 0, 1.364. (= −2.879). , , 0( ).
TORA, x1 = 59.909 1 + 0.006 06D1 0, x3 = 5, s3 = 340.909 09 +
0.006 06D1 0, x2 = 29.545 45 + 0.003 03D1 0. , = 0.158,
−9 750 D1 56 250. , 50% (D1 = 5 000 ), .
;
.
1.25 , 0.25 0 .
D3 = 350 − 800 = −450 D3 −400 . .
∆ > 0, x1 = x2 = 2 + ∆3 . 0 < ∆ 3,
∆
r1 + r2 = ∆
3 1 ⇒ . 3 ∆ < 6, r1 + r2 = 3 > 1 ⇒ . ∆ > 6, D1 D2 .
3.6D
2. (a) x1 =A1 , x2 =A2 , x3 =BK .
max z = 80x1 + 70x2 + 60x3
s.t.
x1 + x2 + x3 500, x1 100, 4x1 − 2x2 − 2x3 0
C() 13
x1 , x2 , x3 0
x1 = 166.67, x2 = 333.33, x3 = 0, z = 36 666.67.
(b)
TORA , BK = 10.
10 .
(c) d1 = d2 = d3 = −5 . TROA , x3 : 10 + d2 − d3 0,
s1 : 73.33 + 0.67d2 + 0.33d1 0,
s3 : 1.67 − 0.17d2 + 0.17d1 0,
.
5. (a) xi = i .
max z = 60x1 + 40x2 + 25x3 + 30x4
s.t.
8x1 + 5x2 + 4x3 + 6x4 8 000, x1 500, x2 500,
x3 800, x4 750, x1 , x2 , x3 0
x1 = 500, x2 = 500, x3 = 375, x4 = 0, z = 59 375 .
(b)
TORA , 8.75 + d2 0. 2 8.75 .
(c) d1 = −15 , d2 = −10 , d3 = −6.25 , d4 = −7.50 . TORA ,
x4 : 7.5 + 1.5d3 − d4 0,
s1 : 6.25 + 0.25d3 0,
s2 : 10 − 2d3 + d1 0,
s3 : 8.75 − 1.25d3 + d2 0,
, z 25%.
(d) x4 = 7.5, 7.50 .
3.6E
5. , x1A + x1B 100 .
9. (a) A 10.27 . 12 , A.
(b) B 0 . , .
5.10 .
14
C() 4
4.1A
2. y1 , y2 , y3 .
max w = 3y1 + 5y2 + 4y3
y1 + 2y2 + 3y3 15, 2y1 − 4y2 + y3 12
s.t.
y1 0, y2 0, y3
4. (c) y1 y2 .
min w = 5y1 + 6y2
s.t.
2y1 + 3y2 = 1, y1 − y2 = 1
y1 , y2
5. y2 −M .
−∞, y2
.
, M → ∞ ⇒ y 4.2A
.
(e) V 2 A = (−14 − 32).
1. (a) AV 1
4.2B
1. (a)
⎛
1
4
− 12
⎜ 1
⎜ −
⎜
= ⎜ 8
⎜ 3
⎝ 8
⎞
− 54
⎟
0 0 ⎟
⎟
⎟
1 0 ⎟
⎠
− 34
0 1
3
4
1
8
0 0
4.2C
3. y1 y2 .
min w = 30y1 + 40y2
s.t.
y1 + y2 5, 5y1 − 5y2 2, 2y1 − 6y2 3
y1 −M (⇒ y1
), y2 0
y1 = 5, y2 = 0, w = 150.
6. y1 y2 .
min
w = 3y1 + 4y2
s.t.
y1 + 2y2 1, 2y1 − y2 5, y1 3
y2
C() 15
y1 = 3, y2 = −1, w = 5.
8. (a) (x1 , x2 ) = (3, 0), z = 15, (y1 , y2 ) = (3, 1), w = 14.
= (14, 15).
9. (a) , , z = w = 17.
4.2D
2. (a) (x2 , x4 ) = (3, 15) ⇒ .
(x1 , x3 ) = (0, 2) ⇒ .
4.
z
x1
x2
x3
x4
x5
0
0
− 25
− 51
0
−1
1
1
12
5
3
5
6
5
x1
1
0
x2
0
1
x5
0
0
− 35
4
5
1
5
− 53
0
0
0
, .
7. , z = c1 x1 + c2 x2 ; , w = b1 y1 + b2 y2 +
b3 y3 . b1 = 4, b2 = 6, b3 = 8, c1 = 2, c2 = 5 ⇒ z = w = 34.
4.3A
2. (a) (x1 , x2 , x3 , x4 ) =SC320, SC325, SC340 SC370 max
z = 9.4x1 + 10.8x2 + 8.75x3 + 7.8x4
s.t.
10.5x1 + 9.3x2 + 11.6x3 + 8.2x4 4 800
20.4x1 + 24.6x2 + 17.7x3 + 26.5x4 9 600
3.2x1 + 2.5x2 + 3.6x3 + 5.5x4 4 700
5x1 + 5x2 + 5x3 + 5x4 4 500
x1 100, x2 100, x3 100, x4 100
(b) , (= 0.494 4).
(c) 0(−0.684 7, −1.361, 0 −5.300 3), .
(d) 0.494 4 /, (8 920, 10 201.72), 6.26%.
4.3B
, = −2.
3.
PP3 PP4 . 0.142 9 1.142 9.
, PP3, 0.142 9 , PP4 1.142 9 .
2.
16
C() 4.4A
1. (b) . E , .
4. (c) x1 M . .
4.5A
4. Q (= 5 200, 9 600, 15 000, 20 000, 26 000, 32 000, 38 000,
42 000, = 1, 2, · · · , 8). = 0.028Q, =
0.649Q, = 0.323Q. = 0.812 21Q.
4.5B
1. (a) .
4.5C
= 12 , 0, 0, 0 . .
1 11
(c)
= − 8 , 4 , 0, 0 . z − 0.125s1 + 2.75s2 = 13.5.
x2 = 2, x3 = 4, z = 14.
2. (a)
x1 = 2,
4.5D
1.
p
100 (y1
3. (a)
+ 3y2 + y3 ) − 3 0. y1 = 1, y2 = 2 y3 = 0, p 42.86%.
= 3y1 + 2y2 + 4y3 − 5 = 2 > 0. ,
.
5
5.1A
4. ,
M .
6. (a b) M = 10 000. , =$49 710.
1
1
2
(c) 1 600
23
320
500
3
4
2
13
700
17
25
1000
36
=$13 000.
400
25
300
350
480
450
5
1000
42
3
M
30
25
40
30
13
C() 9. (100 ).
2
A1
1
2
3
200 . =$304 000.
A2
12
4
A3
18
M
10
8
2
6
30
4
1
20
5
25
12
6
6
M
4
50
50
2
2
4
8
7
5.2A
2. =$804. .
2
3
24
6
6
18
0
12
12
0
0
0
2
14
0
0
0
0
0
20
0
0
0
14
0
0
4
0
2
0
0
12
0
0
0
0
22
5. =$190 040. .
1
500
500
1 400, 2 100
2
600
600
2 200, 3 220, 3
200
200
3 200
4
300
200
4 200
4 180
5.3A
3. (a) =$42. =$37. Vogel
=$37.
5.3B
5. (a) =$1 475.
(b) c12 3, c13 8, c23 13, c31 7.
5.4A
5. (, ) 17
. ,
(D,3)–(A,7)
18
C() 63
, 6 7 =$1 180. .
(A,7)
(D,3)
(D,10)
(D,17)
(D,25)
6. (A,12)
400
300
300
300
300
400
300
300
, $400. (A,21)
,
(A,28)
300
300
400
300
280
300
300
400
I-d,II-c,III-a,IV-b.
5.5A
4. =$1 550. , 1
1
2
2
50
50
0
200
.
3
0
50
6
6.1A
1. (i); (a)1-3-4-2; (b)1-5-4-3-1; (c d) C.5.
C.5
4. , . 1 , . . C.6.
8
C.6
C() 19
6.2A
2. (a) 1-2,2-5,5-6,6-4,4-3. =14
.
5. 1-2-3-4-6. 1-5-7 5-9-8.
6.3A
1. 1 4 . =8 900 .
C.7.
C.7
4. (i, vi ) − (i + 1, vi+1 ), p(q) =
1 2. ( i ). 80 .
C.8.
C.8
6.3B
1. (c)
4,5,6,7,8, 4-6-8.
6.3C
1. (a) 5-4-2-1, =12.
. 8, 4-5-6-8 20
4.
C() C.9. , . Bob-Kay-Rae-Kim-Joe. 4.
Bob Joe C.9
6.3D
1. (a) 1 5 1-3-5 1-3-4-5, 90.
1 −1, 0. 6.4A
1. 11-2, 1-4, 3-4, 3-5,
60.
6.4B
1. (a) (2-3)=40, (2-5)=10, (4-3)=5.
(b) 220 . 330 . 420 .
(c) . 1 .
4. Rif–3, Mai–1, Ben–2, Kim–5. Ken .
7. 6.5A
3.
C.10.
C.10
C() 6.5B
1. 1-3-4-5-6-7.
19.
6.5C
3. (a) 10; (b) 5; (c) 0.
5. (a) 1-3-6,
45 .
(b) A, D, E.
(c) C, D, G 5 . E (d) 2
.
.
7
7.1A
−
+
1. G5 min s+
5 , 55xp + 3.5xf + 5.5xs − 0.067 5xg + s5 − s5 = 0.
3. x1 , x2 , x3 .
Gi : min
s.t.
s−
i , i = 1, 2, · · · , 5
+
x1 + x2 + x3 + s−
1 − s1 = 1 200
+
2x1 + x2 − 2x3 + s−
2 − s2 = 0
+
−0.1x1 − 0.1x2 + 0.9x3 + s−
3 − s3 = 0
+
0.125x1 − 0.05x2 − 0.556x3 + s−
4 − s4 = 0
+
−0.2x1 + 0.8x2 − 0.2x3 + s−
5 − s5 = 0
5. xj j min
s.t.
, j = 1, 2, 3.
+
z = s−
1 + s1
+
−100x1 + 40x2 − 80x3 + s−
1 − s1 = 0
4 x1 5, 10 x2 20, 3 x3 20
7.2A
−
−
+
+
1. min z = s−
1 + s2 + s3 + s4 + s5
xp = 0.020 1, xf = 0.045 7, xs = 0.058 2, xg = 2 , s+
5 = 1.45
145 .
4. x1 , x2 , x3 .
+
−
−
+
min z = s−
1 + s2 + s3 + s4 + s5
21
22
C() x1 = 166.08 , x2 = 2 778.56 , x3 = 3 055.36 , z = 0. .
, 3 4 .
7. j xj , j = 1, 2.
.
−
+
+
min z = 100s−
1 + 100s2 + s3 + s4
+
x1 = 80, x2 = 60, s+
3 = 100 , s4 = 120 .
1
100 , 2
120 , .
7.2B
2. G1 xp = 0.017 45, xf = 0.045 7, xs = 0.058 2, xg = 21.33, s+
4 = 19.33, 0. G1 , G2 G3
, G4 .
−
−
G4 G1 , s1 = 0, s2 = 0, s−
3 = 0.
G4 xp = 0.020 1, xf = 0.045 7, xs = 0.058 2, xg = 2, s+
5 = 1.45. 0. G5 .
G5 G4 , s+
4 = 0.
(s+
G5 G4 , G5 5 = 1.45).
8
8.1A
3. xij
i=2
j i , i = 1
, i = 3
.
x11 + x12 + x13 = 7, x21 + x22 + x23 = 7, x31 + x32 + x33 = 7
x11 + 0.5x21 = 3.5, x12 + 0.5x22 = 3.5, x13 + 0.5x23 = 3.5
x11 + x21 + x31 = 7, x12 + x22 + x32 = 7, x13 + x23 + x33 = 7
.
.
1
2
3
1
3
3
5
1
1
1
3
3
,
C() . xj
j .
6. y
x4
23
, j = 1, 2, 3.
min z = y
s.t.
3x1 − y = 2, x1 + 3x2 − y = 2, x1 + x2 + 3x3 − y = 2
y − x1 − x2 − x3 − 3x4 = 1
y = 79 + 81n, n = 0, 1, 2, · · ·
10. 5,6 8( 27 ). 1,2,3,4 7( 28 ). .
12.
i j,
xij = 1; 0. cij
j .
, Cj
max z =
s.t.
6
j=1
10
10 6
cij xij
i=1 j=1
xij = 2, i = 1, 2, · · · , 10
xij Cj , j = 1, 2, · · · , 6
i=1
1, (2,4,9); 2,(2,8); 3,(5,6,7,9);
5,(1,3,8,10); 6,(1,3). 1 775.
4,(4,5,7,10);
8.1B
1.
j,
xj = 1; xj = 0. 10 + 32 + 4 + 15 + 9 = 80( ).
(ABC,1,2,3,4,ABC) min z = 80x1 + 50x2 + 70x3 + 52x4 + 60x5 + 44x6
s.t.
x1 + x3 + x5 + x6 1, x1 + x3 + x4 + x5 1, x1 + x2 + x4 + x6 1,
x1 + x2 + x5 1, x2 + x3 + x4 + x6 1,
j, xj = (0, 1).
(1,4,2) (1,3,5), z = 104.
1 .
2. a, d, f . .
t,
xt = 1; xt = 0. c ,
xc = 1;
7.
xc = 0. ct
t . Sc
c j .
. Pj
24
C() max z =
s.t.
15
Pc xc
c=1
xt xc ,
c = 1, 2, · · · , 15,
7
ct xt 15
t=1
t∈Sc
2,4,5,6 7. 1
, .
8.1C
j , j = 1, 2, 3.
2. xj
j,
yj = 1; yj = 0.
min
z = 2x1 + 10x2 + 5x3 + 300y1 + 100y2 + 200y3
s.t.
x1 + x2 + x3 2 000, x1 − 600y1 0, x2 − 800y2 0,
x3 − 1 200y3 0, x1 500, x2 500, x3 500
x1 , x2 , x3 , y1 , y2 , y3 = (0, 1).
x1 = 600, x2 = 500, x3 = 900, z = 11 300().
3. 1 1 2, 2 3 4. z = 18.
10. xe
Eastern (), xu
US Air
Continental ().
(), xc
e1 e2 , u c .
max
z = 1 000(xe + 1.5xu + 1.8xc + 5e1 + 5e2 + 10u + 7c)
s.t.
e1 xe /2, e2 xe /6, u xu /6, c xc /5, xe + xu + xc = 12.
2 Eastern
39 000 .
, 10 Continental
. 8.1D
1. xij
0.
(i, j) . , 3
3
xij = 15, i = 1, 2, 3,
xij = 15, j = 1, 2, 3
j=1
i=1
x11 + x22 + x33 = 15, x31 + x22 + x13 = 15
x11 x12 + 1x11 x12 − 1
x11 x13 + 1x11 x13 − 1
x12 x13 + 1x12 x13 − 1
x11 x21 + 1x11 x21 − 1
C() 25
x11 x31 + 1x11 x31 − 1
x21 x31 + 1x21 x31 − 1
i j, xij = 1, 2, · · · , 9
2
9
4
7
5
3
6
1
8
3. xj
j .
.
max z = 25x1 + 30x2 + 22x3
s.t.
3x1 + 4x2 + 5x3 100
4x1 + 3x2 + 6x3 100
x1 , x2 , x3 0
3x1 + 4x2 + 5x3 90
4x1 + 3x2 + 6x3 120
.
26 1, 3 2, 3. 2 .
8.2A
2. (a) z = 6, x1 = 2, x2 = 0;
(d) z = 12, x1 = 0, x2 = 3.
3. (a) z = 7.25, x1 = 1.75, x2 = 1;
(b) z = 10.5, x1 = 0.5, x2 = 2.
9. 0 − 1 ILP max
z = 18y11 + 36y12 + 14y21 + 28y22 + 8y31 + 16y32 + 32y33
s.t.
15y11 + 30y12 + 12y21 + 24y22 + 7y31 + 14y32 + 28y33 43
z = 50, y12 = 1, y21 = 1, 0. , x1 = 2, x2 = 1.
0 − 1 41 , 29 .
8.2B
1. (a) , , . LP .
6. (a) (x1 , x2 , x3 ) = (2, 1, 6), z = 26. (x1 , x2 , x3 ) =
(3, 1, 6), .
TORA B&B .
26
C() 8.3A
1. i j , . “”.
1
2
3
4
5
6
1
−
4
4
6
6
5
2
4
−
6
4
6
3
3
4
6
−
4
8
7
4
6
4
4
−
6
5
5
6
6
8
6
−
5
6
5
3
7
5
5
−
8.3C
2. C.11.
C.11
9
9.1A
1. =21
. 1-3-5-7.
C() 27
9.2A
3. =17
. 1-2-3-5-7.
9.3A
2. (a) =120. (m1 , m2 , m3 ) = (0, 0, 3), (0, 4, 1), (0, 2, 2) (0, 6, 0).
5. =250.
I 2
,
II 3
,
,
IV 1
.
xj = 1; 0. 7. j ,
III 4
max
z = 78x1 + 64x2 + 68x3 + 62x4 + 85x5
s.t.
7x1 + 4x2 + 6x3 + 5x4 + 8x5 23, xj = (0, 1), j = 1, 2, · · · , 5
. =279.
9.3B
1. (a) 1 6 , 2 1 , 3 2 , 4 3 , 5 2 .
3. 1 7 , 2 3 , 3 4 , 4 .
9.3C
2. 4 , , , . =$458.
9.3D
3. (a) xi , yi i , zi = xi + yi .
fn (zn ) = max {pn yn }
yn =zn
fi (zi ) = max{pi yi + f
i+1
yi zi
(2zi − 2yi )}, i = 1, 2, · · · , n − 1
10 10.3A
2. (a) = $51.50;
(b) = $50.20, y ∗ = 239.05 .
4. (a) 1, $2.17,
(b) 10 ,
2 $2.50.
100 .
28
C() 10.3B
2. 130 ,
500 . = $258.50.
. , 4. T CU1 (ym ) T CU2 (q),
0.934 4% .
10.3C
1. AMPL/Solver (y1 , y2 , y3 , y4 , y5 ) = (4.42, 6.78, 4.12, 7.2, 5.8), =$568.12.
4.
4
i=1
365Di
yi
150.
Solver/AMPL (y1 , y2 , y3 , y4 ) = (155.3, 118.82, 74.36, 90.09), =$54.71.
10.4A
1. (a) 1, 4, 7, 10
,
500 .
10.4B
3. 1 173 , , 2 180 , 3 240 , 4 110
5 203 .
10.4C
1. (a) , .
(b) (i) 0 z1 5, 1 z2 5, 0 z3 4, x1 = 4, 1 x2 6, 0 x3 4.
(ii) 5 z1 14, 0 z2 9, 0 z3 5, x1 = 0, 0 x2 9, 0 x3 5.
2. (a) z1 = 7, z2 = 0, z3 = 6, z4 = 0. =$33.
10.4D
1. 1 2 4 , , 4 0, 22, 90, 67. 2 112 , 4 67 . =$632.
10.4E
1. 1 210 , 4 255 , 7 210 , 10
165 .
11 11.1A
1. A, B, C =(0.442 14, 0.251 84, 0.306 02)
C() 29
11.1B
2. A
, CR > 0.1. (wS , wJ , wM ) = (0.331, 0.292, 0.377). Maisa.
4. . (wH , wP ) = (0.502, 0.489). H.
11.2A
2. (a) C.12;
(b) EV ()=−$8 250, EV ()=$250. .
6. (a) C.13;
C.12
(b) EV ()=−$0.025. C.13
.
30
C() 12. =8 . =$397.50.
=49 /.
19. 99∼151 .
15. 11.2B
2. z 5 1 , P{A|z} = 0.607 9,
P{B|z} = 0.390 3.
4. (a) (b)
7. (b)
=$196 000. , =$163 000.
, , .
, B, A.
,
11.2C
=$5, .
(b) 0 x < 10, U (x) = 0; x = 10, U (x) = 100.
(c) .
1. (a)
11.3A
1. (a) (
(b) a1 );
a2 a3 .
11.4A
(2, 3). 3. (a) 2 < v < 4.
2. (a)
=4.
11.4B
1. 50:50 , 2. =0.
100%A
50%A:50%B
A
100
50
100%B
0
B
0
30
100
50:50 100%A 100%B.
Robin 50:50 A B. =$50(=Robin ).
11.4C
1. (a) 1 C() AB
AC
AD
BC
BD
CD
AB
1
0
0
0
0
−1
AC
0
1
0
0
−1
0
AD
0
0
1
−1
0
0
BC
0
0
−1
1
0
0
BD
0
−1
0
0
1
0
CD
−1
0
0
0
0
1
31
50:50 AB CD. =0.
3. (a) (m, n) =( ! 1 , ! 2 ). , ! 1;
, −1. Botto (1, 1) " (0, 3),
! 1 ! 2, 1 + (−1) = 0. Blotto 3,0
2,1
1,2
0,3
2,0
−1
−1
0
0
1,1
0
−1
−1
0
0,2
0
0
−1
−1
Blotto Blotto 50:50 (2, 0) (0, 2),
(3, 0) (1, 2). " 50:50
=−0.5, Blotto . .
12 12.1A
1. (a) =71%;
(b) .
12.2A
1.
(a)
#
(b)
(h)
12.3A
1. (b) (i) 6 , = 16 .
/, =0.2 .
3. (a) f (t) = 20e−20t , t > 0;
(b) P t > 15
60 = 0.006 74.
7. Jim 2 P{t 1} = 0.486 6, 2 P{t 1} = 0.513 4. 8 , Jim Ann=17.15 .
10. (a) P{t 4} = 0.486 6;
(b) =6.208.
(c) µ = 5
32
C() 12.4A
1. pn5 (1) = 0.559 51.
4. (a)p2 (t = 7) = 0.241 67.
6. (a) λ =
1
10
+ 17 , p2 (t = 5) = 0.219.
12.4B
2. (a) p0 (t = 3) = 0.005 32;
(b) pn17 (t = 1) = 0.950 2.
5. p0 (4) = 0.371 16.
8. (a) =25 − 7.11 = 17.89 ;
(b) p0 (t = 4) = 0.000 69.
12.5A
3. (a) pn3 = 0.444 5;
(b) pn2 = 0.555 5.
6. (a) pj = 0.2, j = 0, 1, 2, 3, 4;
(b) =2 ;
(c) p4 = 0.2.
12.6A
1. (a) Lq = p6 + 2p7 + 3p8 = 0.191 7 .
(b) λlost = 0.126 3 /, 8 (c) =c − (Ls − Lq ) = c −
8
n=0
npn +
=1.01 .
8
(n − c)pn .
n=c+1
12.6B
2. (a) p0 = 0.2;
=$40 × Lq = $128.
5. (a) p0 = 0.4;
(c) Wq = 2.25 ;
6. (d) 13 .
(b) =$50 × µt = $375;
(c)
(b) Lq = 0.9 ;
(d) pn11 = 0.003 6.
12.6C
1. P{τ > 1} = 0.659.
5. $37.95/.
12.6D
1. (a) p0 = 0.365 4;
(b) Wq = 0.207 ;
= 4 − Lq = 3.212;
(d) p5 = 0.048 12.
(e) Ws 40%, 9.6 (µ = 10 /).
(c)
C() 33
(b) Lq = 6.34 ;
, 0.4. 60%.
7. (a) 1 − p5 = 0.962;
(b) λlost = λp5 = 0.19 /.
4. (a) p8 = 0.6;
(c) 12.6E
2. c = 2, Wq = 3.446 ; c = 4, Wq = 1.681 , 51%.
5. K $, TORA , p0 + p1 + · · · + pK+2 0.999, K 10.
7. (a) pn4 = 0.657 72;
(b) =0.667
.
12.6F
2. (c) =81.8%;
(d) p2 + p3 + p4 = 0.545.
4. (a) p40 = 0.000 14;
(b) p30 + p31 + L + p39 = 0.024 53;
# =Ls − Lq = 20.043 − 0.046 ≈ 20;
(f) =1 − pn29 = 0.024 67. 8 4.
(d)
12.6G
2. (a) 7 ;
(b)pn8 = 0.291 1.
12.6H
1. (b) =2.01.
} = p0 + p1 = 0.344 92.
4. (a) Ls = 1.25
;
(b) p0 = 0.333 42;
(c) Ws = 0.25 .
6. λ = 2 /( · ), µ = 0.5 /, R = 5, K = 5.
(a) =5 − Ls = 1 .
(b) p5 = 0.327 68.
(c) pn2 = 0.057 92.
(d) P{23
12.7A
2. (a) E{t} = 14 , var{t}=12 2 . Ls = 7.867 2 .
4. λ = 0.062 5 /, E{t} = 15 , var{t} = 9.33 2 .
(a) p0 = 0.062 5;
(c) Ws = 132.17 .
(b) Lq = 7.3 ;
34
C() 12.9A
2. (M/M/1) : (GD/10/10). $386.50.
4. (b) µ = λ +
c2 λ
c1 ;
1 $431.50, 2 (c) =2 725 /.
12.9B
2. (a) 2 $86.4, 3 $94.80.
(b) 2 , =$30 × Ws = $121.11; 3 $94.62.
4. λ = 0.361 25/, µ = 10/. (M/M/1) : (GD/20/20)
Ls = 0.705 29
. =$36.60, 3 =$60.
12.9C
1. (a) 5;
(b) 4.
C ()
13 13.1A
2. (1, 0) (0, 2) Q , 0 < λ < 1, λ(1, 0) + (1 − λ)(0, 2) = (λ, 2 − 2λ) Q .
13.1B
2. (b) , x1 > 1, 0 < x2 < 1. C.14.
(d) .
(f ) .
C.14
3. (a) , det(B) = −4.
(d) , 3 .
13.1C
1.
B
−1
=
0.3
−0.2
0.1
0.1
x1
x2
x3
x4
z
1.5
−0.5
0
0
21.5
x3
x4
0
0.5
0.5
0
1
0
0
1
2
1.5
, .
4. z = 34.
max z = 2x1 + 5x2 s.t. x1 4, x2 6, x1 + x2 8, x1 , x2 0.
, * .
2
C() 13.2A
1. (a) P 1 .
(b) B = (P 2 , P 4 ) .
2. X B , {zj − cj } = cB B −1 B − cB = cB I − cB = cB − cB = 0
7. , 10. , 11. (a) xj =
1
,
α
n − m.
, xj .
(b) xj =
β
,
α
.
xj .
13.2B
2. (b) (x1 , x2 , x3 ) = (1.5, 2, 0), z = 5.
13.3A
2. (b) (x1 , x2 , x3 , x4 , x5 , x6 ) = (0, 1, 0.75, 1, 0, 1), z = 22.
13.4A
2. max w = Y b s.t. Y A c, Y 0.
13.4B
5. 1: (b1 , b2 , b3 ) = (4, 6, 8) ⇒ 2: (c1 , c2 ) = (2, 5) ⇒ = 34.
= 34.
7. min w = Y b s.t. Y A = C, Y .
13.5A
1. − 27 t < 1.
2. (a)
t 0t
(x2 , x3 , x1 ) = ( 25
,
4
1
3
5
2
(x2 , x4 , x1 ) =
5. {zj − cj }j=1,4,5 = 4 −
(x2 , x3 , x6 ) = (5, 30, 10)
3t
2
−
3t2
,1
2
90
, 5)
4
5
( 2 , 15, 20)
− t2 , 2 −
t
2
+
t2
2
t 1
3
5
2
t < ∞
. 0 t 1, .
13.5B
1. (a) t1 = 10, B 1 = (P 2 , P 3 , P 4 ).
2. t = 0 , (x1 , x2 , x4 ) = (0.4, 1.8, 1). 0 t 1.5, .
. t > 1.5, C() 3
14 14.1A
1. (a) 0.15 0.25. (b) 0.571. (c) 0.821.
2. n 23.
3. n > 253.
14.1B
3.
5
.
32
4. p =Liz , John 3p, 6p. 4 3
.
13
(a)
(b)
7
.
13
(c)
Jim . Ann , p + 3p + 3p + 3p + 6p = 1.
6
.
13
14.1C
3. (a) 0.375. (b) 0.6.
7. 0.954 5.
14.2A
2. (a) K = 20.
3. P{ 1 100} = 0.3.
14.3A
3. (a) P{50 (b)
(c) 70} = 0.666 7.
= 2.67.
= $22.33.
14.3B
1. =3.667 =1.556.
14.3C
1. (a) P(x1 = 1) =P(x2 = 1) = 0.4, P(x1 = 2) = P(x2 = 2) = 0.2, P(x1 = 3) =P(x2 =
3) = 0.4.
(b) 14.4A
1.
1 10
2
.
2. 0.054 7.
P(x1 , x2 ) =P(x1 )P(x2 ).
4
C() 14.4B
1. 0.864 6.
3. (a) P{n = 0} = 0.
(b) P{n 3}; 1.
14.4C
1. λ = 12 /. P{t 5} = 0.63.
14.4D
2. 0.001 435.
15 15.1A
1. (a) 537 , 1 000 .
15.1B
2. y ∗ = 317.82 , R∗ = 46.82 .
3. y ∗ = 316.85 , R∗ = 58.73 . 15.1-2 , y ∗ = 319.44 , R∗ = 93.61 .
15.1-2 , R∗ , .
15.2A
3. 0.43 p 0.82.
6. 32 .
15.2B
1. x < 4.53, 9 − x , .
15.3A
2. x < 4.61, 4.16 − x , .
16 16.1A
1. (a) P{H} =P{T } = 0.5. 0 R 0.5, Jim
10 0.5 R 1, Jan
10 .
7. 0 R 0.5, L = 1 0.5 R 1, L = 2 . 0 R 0.2, = 0 0.2 R 0.9, = 1 ; 0.9 R 1, = 2 .
R L. L = 1
,
C() R , L = 2, , 5
,
.
16.2A
1. (a) .
16.3A
4. C.15.
C.15
16.3B
1. t = − λ1 ln(1 − R), λ = 4 .
R
t( )
1
2
3
4
–
0.058 9
0.673 3
0.479 9
–
0.015 176
0.279 678
0.163 434
0
0.015 176
0.294 855
0.458 288
2. t = a + (b − a)R.
4. (a) 0 R 0.2: d = 0; 0.2 R 0.5: d = 1; 0.5 R 0.9: d = 2; 0.9 R 1:
d = 3;
9. 0 R p, x = 0; x =(
ln(1−R)
ln q
).
16.3C
1. y = − 15 ln(0.058 9 × 0.673 3 × 0.479 9 × 0.948 6) = 0.803 .
2. t = x1 + x2 + x3 + x4 , xi = 10 + 10Ri , i = 1, 2, 3, 4.
16.4A
1. 16.4-1 ,
4. , 50 , 16.5A
2. (a) .
3. (a) 1.48 .
(b) .
(b) 7.4 .
.
C() 6
16.6A
2. 15.07 µ 23.27.
17 17.1A
2. S1:
S2: S3:
S4:
S5: S1
S2
S1
0.4
0.1
S2
0.6
0.3
S3
0
0.6
S4
0
0
S5
0
0
S3
S4
S5
0.1
0.4
1
0
0
0
0.5
0
0
0.4
0
0
0
0.6
0
S1
S2
S3
S4
S5
0
0
1
0
0
17.2A
2.
(2 2
S1
S2
S3
S4
S5
S1
S2
S3
0.4
0.1
0.1
0.6
0.3
0
0
0.6
0.5
0
0
0.4
0
0
0
S4
S5
0.4
1
0
0
0
0
0
0
0.6
0
) (P 2 )
S1
S1
0.22
S2
0.42
S3
0.36
S4
0
S5
0
S2
S3
0.13
0.25
0.15
0.06
0.48
0.25
0.24
0.2
0
0.24
S4
S5
0.76
0.4
0.24
0.6
0
0
0
0
0
0
2 = (0 0 1 0 0)P 2
C() P{ , S4, 2 (2 )
S1
S2
S3
0.25
0.06
0.25
S4
S5
0.2
0.24
}=0.2.
17.3A
excelMarkovChains.xls, 1. (a)
(b) 1, 2, 3 , 4
, 3.
.
17.4A
1. (a)
S
S
0.8
C
0.2
R
0
C
R
0.3
0.1
0.5
0.1
0.2
0.8
(π1 , π2 , π3 ) = (π1 , π2 , π3 )P
π1 + π2 + π3 = 1
S
C
R
2.0
4.0
4.0
=2 × 0.5 + 1.6 × 0.25 + 0.4 × 0.25 = $1 500
(b) µSS = 2 5. (a)
0.50
0.25
0.25
, .
0.95
0.06
0.04
0.9
0.01
0.04
0
0.1
0.9
(b)
44.12% 0.441 175
0.367 646
2.266 672 8
2.720 008 9
0.191 176
5.230 789 2
, 36.76% , 19.11% .
7
8
C() (c) =0.12($5 000 × 0.367 6 + 12 000 × 0.191 1) × 70 000 000
= $34 711 641 097.07
14. (a) =(i, j, k)=( −2
i, j, k = (0
, , −1
, 1)
, (1-0-0) (0-0-1).
0-0-0
1-0-0
0-1-0
0-0-1
1-1-0
1-0-1
0-1-1
1-1-1
0-0-0
1-0-0
0-1-0
0.1
0.2
0
0
0
0.2
0
0
0
0.9
0.8
0
0
0
0
0
0
0.8
0
0
0
0
0
0
0-0-1
1-1-0
0
0
0
0.3
0.2
0
0
0
0
0
0
0.7
0.8
0
0
0
1-0-1
0-1-1
1-1-1
0
0
0
0
0
0
0.3
0
0
0
0
0
0
0.3
0.5
0
0
0
0.7
0
0
0
0.7
0.5
(b)
3 0-0-0
1-0-0
0.014 859
0.066 865
0-1-0
0-0-1
0.066 865
0.066 865
1-1-0
1-0-1
0-1-1
0.178 306
0.178 306
0.178 306
1-1-1
0.249 629
= 1(0.066 865 + 0.066 865 + 0.066 865)
+2(0.178 306 + 0.178 306 + 0.178 306)
+3(0.249 629) = 2.019 32
= 2.019 32/3 = 0.673 11
17.5A
1. (a)
1
2
3
4
5
1
0
0
0
0
:
1
0
2
0.333 3
3
0.333 3
4
0.333 3
0
5
0.333 3
0.333 3
0.5
0
0.333 3
0
0.333 3
0
0
0
0
0
0.333 3
0.333 3
0.5
0
0.333 3
0.333 3
0.333 3
0
),
C() (3 )
1
2
3
0.074 07
0.296 3
0.296 3
0.214 286
0.214 286
0.214 286
4
5
0.259 26
0.074 07
0.142 857
0.214 286
(b) a5 = 0.074 07
(c) π5 = 0.214 286
(d) µ5 = 4.666 6
5. (a)
(I − N)−1
Mu
1
2
1
2
1
2
1
1.625
3
1
0.875
5
0.666 7
0.333 3
4.666 6
3.833 3
3
4
1
1
0.875
0.5
1.625
0.5
0.333 3
1.333 3
3.833 3
3.333 3
A
B
A
0.75
0.2
B
0.1
0.75
C
0.15
0.05
C
0.125
0.125
0.75
(b)
A
0.394 737
B
C
0.307 018
0.298 246
A: 39.5%, B:30.7%, C:29.8%
(c)
(I − N)−1
A
C
C
3.428 57
5.714 29
A
C
B
9.142 86
8.571 43
A
1
5.882 35
2
2.352 94
A
C
8.235 29
B
4.705 88
5.882 35
B
1.588 2
A→B: 9.14 A→C: 8.23 17.6A
2. (a) Mu
A
5.714 29
2.857 14
1 , 2 , 3 , 9
10
C() P
1
2
1
0
0
2
0.3
0
3
0
0.1
0.7
0.9
3
0
0
0
0
0
0
1
1
(b)
(I − N)−1
2
1
Mu
3
1
2
1
0
0.3
1
0.03
0.01
1
2
1.33
1.1
3
0
0
1
3
1
1.33 .
8. (a)
1
2
P
3
4
1
2
3
0.2
0
0
0.8
0.22
0
0
0.78
0.25
0
0
0.75
0
0
0
4
F
0
0
0
0
0
0
0.3
0
0.7
1
(b)
(c) 1
1
1.25
(I − N)−1
2
3
1.282
1.333
4
1.429
1
Mu
F
5.29
2
3
0
0
1.282
0
1.333
1.333
1.429
1.429
2
3
4.04
2.76
4
0
0
0
1.429
4
1.43
=5.29, (d) 10. (a) F
, 16 (4 ) , (c).
0,1,2,3,D(
)
. .
P
(b) 0
1
0
0.5
0.4
1
0.5
0
2
0
0.6
3
0
0
D
0
0
2
3
D
0.3
0.2
0
0
0
0
0
0
0
0.7
0
0
0
0.8
1
12 .
C() (I − N)−1
11
Mu
0
1
0
5.952
3.952
1
2.976
2.976
2
1.786
1.786
3
1.25
1.25
0
1
D
12
9.96
2
3
2.619
1.19
1.31
0.595
1.786
0.357
1.25
1.25
2
3
6.96
3.39
(c) 6.96 .
18 18.1A
1. (a) .
(b) x = 0 (e) x = 0 .
, x = 0.63 4. (x1 , x2 ) = (−1, 1)
, x = −0.63 .
(2, 4).
18.2A
1. (b) (∂x1 , ∂x2 ) = (2.83, −2.5)∂x2 .
18.2B
3. 2(xi −
√
n
∂f = 2δ C 2−n .
x2
n
)
xi
= 0, i = 1, 2, · · · , n − 1. xi =
5
6. (b) (x1 , x2 , x3 , x4 ) = (− 74
, − 10
,
74
155 60
, ),
74 74
√
n
C, i = 1, 2, · · · , n.
.
18.2C
2. (x1 , x2 , x3 ) = (−14.4, 4.56, −1.44) (4.4, 0.44, 0.44).
19 19.1A
2. (c) x = 2.5, (e) x = 2, ∆=0.000 001.
∆=0.000 001.
19.1B
1. , ∇f (X ) = ∇f (X 0 ) + H (X − X 0 ). Hessie
f (X ) .
0 , X = X 0 − H −1 ∇f (X 0 ). X
, X .
H
X , , , . , ∇f (X ) =
∇f (X) = 0 , X 0 12
C() 19.2A
2. x1 = 0, x2 = 3, z = 17.
4. wj = xj + 1, j = 1, 2, 3,v1 = w1 w2 , v2 = w1 w3 , max z = v1 + v2 − 2w1 − w2 + 1
s.t.
v1 + v2 − 2w1 − w2 9
ln v1 − ln w1 − ln w2 = 0
ln v2 − ln w1 − ln w3 = 0
19.2B
1. x1 = 1, x2 = 0, z = 4.
2. x1 = 0, x2 = 4, x3 = 0.7, z = −2.35.
19.2C
1.
max z = x1 + 2x2 + 5x3
s.t.
2x1 + 3x2 + 5x3 + 1.28y 10
9x21 + 16x23 − y 2 = 0
7x1 + 5x2 + x3 12.4
x1 , x2 , x3 , y 0
20 20.1A
1. C.16.
C.16
20.1B
1. 1: .
C() x12
x13
x24
x32
x34
min z
1
5
3
4
6
1
1
1
2
3
4
−1
1
−1
1
−1
0
∞
min z
1
2
3
4
1
−1
−1
2
= 50
30
40
10
∞
10
∞
0
∞
x12
x13
x24
x32
x34
1
5
3
4
6
1
−1
1
1
−1
.
−1
= 20
= −40
1
1
−1
∞
∞
−1
= −40
= 20
= −30
∞
∞
10
= 40
= −20
20.1C
1. = 9 895 . 1 5. = 24 300
210 , 3 . 220 .
.
1
2
1
0
500
2
3
450
0
0
300
1 000
0
0
1 000
1
2
20.2A
1. (c)
x2 M . ,
(x1 , x2 ) = α1 (0, 0) + α2 (10, 0) + α3 (20, 10) + α4 (20, M ) + α5 (0, M )
α1 + α2 + α3 + α4 + α5 = 1, αj 0, j = 1, 2, · · · , 5
, 0) + α3 (0, 12)
2. 1(x1 , x2 ) = α1 (0, 0) + α2 ( 12
5
2(x4 , x5 ) = β1 (5, 0) + β2 (50, 0) + β3 (0, 10) + β4 (0, 5)
α1 = α2 = 0, α3 = 1 ⇒ x1 = 0, x2 = 12
β1 = 0.488 9, β2 = 0.511 1, β3 = β4 = 0 ⇒ x4 = 28, x5 = 0
6. , .
(x1 , x2 , x3 , x4 ) = ( 53 ,
15
, 0, 20),
3
z = 195.
13
14
C() 22 22.1A
2. 1 , . 2 , . 3 .
22.2A
1. 1 , $10 000. 2 , . 3 . 4 .
$35 520
4. 2 1, 3 2, 3 3.
22.3A
3. 1 $1, 2 $1, 3 $1
. =0.109 375.
23 23.1A
2. ,
1
,
,
2
3
2
,
,
3
,
1
2
.
23.2A
1. 1 2 3. , 2 23.3A
1.
1
.
, . 3 .
.
,
1
3
D
D.1
D.1.1
p1 , p2 , · · · , pn n , P ,
P = (p1 , p2 , · · · , pn )
P n (
), P i pi . , P = (1, 2) .
D.1.2
()
n P = (p1 , p2 , · · · , pn )
Q = (q1 , q2 , · · · , qn )
R = (r1 , r2 , · · · , rn )
ri = pi ± qi . , R = P ± Q i D.1.3
P , Q, S, P +Q=Q+P
()
(P ± Q) ± S = P ± (Q ± S)
(
P + (−P ) = 0
(
)
)
P ()θ, Q = θP = (θp1 , θp2 , · · · , θpn )
P
D.1.4
θ
. , P , S θ, γ,
θ(P + S) = θP + θS
()
θ(γP ) = (θγ)P
(
)
P 1 , P 2 , · · · , P n , n
j=1
θj P j = 0 ⇒ θj = 0, j = 1, 2, · · · , n
D 844
n
θj P j = 0,
θj = 0
j=1
. , P 1 = (1, 2), P 2 = (2, 4)
, θ1 = 2 θ2 = −1, θ1 P 1 + θ2 P 2 = 0
D.2
D.2.1
m n A .
m × n ()
⎛
a11
⎜ a21
⎜
A=⎜
⎝ a31
a41
D.2.2
a12
a22
a32
a42
(2) . , (4 × 3) ⎞
a13
a23 ⎟
⎟
⎟ = aij 4×3
a33 ⎠
a43
, m = n.
, (3 × 3) ⎛
1
⎜
I3 = ⎝ 0
0
(3) 1 n .
(4) m 1 .
A
T
AT 1, 0
0
⎛
1
4
1
⎟
0⎠
0
1
⎞
⎟
5 ⎠ ⇒ AT =
6
A = aij , B = bij aij = bij .
0. ⎞
1
2
3
4
5
6
B = 0 (zero matrix), B (7) A (transpose), i j, A aji . ,
⎜
A = ⎝2
3
(6)
i j . (1) (5)
aij aij
.
, , i, j
D.2
845
D.2.3
() .
, (
D.2.6).
()
, A = aij B = bij , (m × n) ,
D = A + B . dij m×n = aij + bij m×n
A, B, C , A+B =B+A
()
A ± (B ± C) = (A ± B) ± C
(
T
T
(A ± B) = A ± B
)
T
A = aij B = bij , A D = AB . A (m × r) , m n , B (r × n) . , D dij =
r
B , , D (m × n)
i j
aik bkj ,
k=1
, A=
D=
=
7
9
2 4
6 8
23 31 9
0
1
34
3
46
5
1
3
2
4
=
, B=
5
7
9
6
8
0
1×5+3×6
1×7+3×8
1×9+3×0
2×5+4×6
2×7+4×8
2×9+4×0
18
, AB = BA, BA .
I m A = AI n = A,
I m , I n (AB)C = A(BC)
C(A ± B) = CA ± CB
(A ± B)C = AC ± BC
αAB = (αA)B = A(αB), α A (m × r) A=
, B (r × n) A11
A12
A13
A21
A22
A23
⎛
B 11
⎜
, B = ⎝ B 21
B 31
B 12
, A B ⎞
⎟
B 22 ⎠
B 32
D 846
, i, j, Aij B ij , A11 B 11 + A12 B 12 + A13 B 31 A11 B 12 + A12 B 22 + A13 B 32
A×B =
A21 B 11 + A22 B 21 + A23 B 31 A21 B 12 + A22 B 22 + A23 B 32
,
⎛
1
⎜
⎝1
2
D.2.4
2
0
5
⎞
⎛
⎞ ⎛
⎞⎛ ⎞
⎛ ⎞
1
4
30
⎟ ⎜
⎟
⎜ (1)(4) + ( 2 3 ) 8
⎟ ⎜ 4 + 2 + 24 ⎟ ⎜ ⎟
⎟⎜ ⎟ ⎜
5⎠⎝1⎠ = ⎜ ⎟ = ⎜ ⎟ = ⎝ 44 ⎠
0 5
1 ⎠ ⎝ 4
40 ⎠
⎝ 1
(4) +
+
6
8
61
2
5 0
8
8
53
3
n ⎛
a11
⎜
⎜ a21
A=⎜
⎜ ..
⎝ .
an1
a12
···
a22
..
.
···
an2
···
a1n
⎞
⎟
a2n ⎟
.. ⎟
⎟
. ⎠
ann
, Pj1 j2 ···jn = a1j1 a2j2 · · · anjn
A ρ n! j1 , j2 , · · · , jn . 1, j1 j2 · · · jn ∈j1 j2 ···jn =
0, j1 j2 · · · jn , A ∈j1 j2 ···jn Pj1 j2 ···jn
ρ
detA
|A|.
⎛
a11
⎜
A = ⎝ a21
a31
a12
a13
⎞
a22
⎟
a23 ⎠
a32
a33
|A| = a11 (a22 a33 − a23 a32 ) − a12 (a21 a33 − a31 a23 ) + a13 (a21 a32 − a22 a31 )
(1) .
, .
T
(2) |A| = |A |.
(3) B A (4) A (
(5) ) α () , |B| = −|A|.
, |A| = 0.
, |A| .
D.2
(6) α, 847
.
(7) A B n , |AB| = |A||B|
|A| aij Mij . , ⎛
a12
a11
⎜
A = ⎝ a21
M11
a22
= a32
Aij = (−1)
A a13
⎟
a23 ⎠
a22
33
a13 ,···
a 31
33
Mij B ||Aij || , ⎛
A11 A21
⎜
A
A22
⎜
12
adj A = Aij T = ⎜
..
⎜ ..
.
⎝ .
, ⎛
A1n
A2n
1
3
⎜
A = ⎝2
3
2
i j ⎞
a31 a32 a33
a
a23 , M22 = 11
a
a i+j
A 2
···
···
···
An1
aij (cofactor),
⎞
⎟
An2 ⎟
.. ⎟
⎟
. ⎠
Ann
⎞
3
⎟
2⎠
3
4
3
A11 = (−1) (3 × 4 − 2 × 3) = 6, A12 = (−1) (2 × 4 − 3 × 2) = −2, · · · ,
⎛
⎞
6
1 −5
⎜
⎟
adj A = ⎝ −2 −5 4 ⎠
−3
D.2.5
3
−1
r, (full-rank)
r. (nonsingular)
⎛
⎞
1 2 3
⎜
⎟
A = ⎝2 3 4⎠
3 5 7
A (singular)
. , |A| = 1 × (21 − 20) − 2 × (14 − 12) + 3 × (10 − 9) = 0
A r = 2, 1
2
2
= −1 = 0
3
D 848
D.2.6
B C n . ,
BC = CB = I, B C , C B
B −1 C −1 .
BC = I, B , C = B −1 , .
BC = I
B −1 BC = B −1 I
IC = B −1
C = B −1
, (1) A B n (2) .
, (AB)−1 = B −1 A−1 .
A , AB = AC
B = C.
n . ⎞⎛
⎞ ⎛ ⎞
⎛
x1
b1
a11 a12 · · · a1n
⎟⎜
⎟ ⎜ ⎟
⎜
⎜ a21 a22 · · · a2n ⎟ ⎜ x2 ⎟ ⎜ b2 ⎟
⎟ ⎜ ⎟
⎜
⎜ .
..
.. ⎟
⎟⎜ . ⎟ = ⎜ . ⎟
⎜ .
.
. ⎠ ⎝ .. ⎠ ⎝ .. ⎠
⎝ .
xn
bn
an1 an2 · · · ann
xi , aij bi . n AX = b
, A . A−1 AX = A−1 b
X = A−1 b
D.2.7
A n A−1 =
,
1
1
adj A =
|A|
|A|
⎛
A11
⎜
⎜ A12
⎜ .
⎜ .
⎝ .
A1n
TORA LU . Press
A21
···
A22
..
.
···
A2n
···
(1986).
An1
⎞
⎟
An2 ⎟
.. ⎟
⎟
. ⎠
Ann
D.2
, ⎛
1
2
⎜
A = ⎝2
⎛
6
⎜
adj A = ⎝ −2
−3
⎛
A−1 =
(–) 6
1 ⎜
⎝ −2
−7
−3
3
1
−5
849
⎞
4
⎞
⎟
4 ⎠ , |A| = −7
−1
⎞
−5
⎛
⎟ ⎜
4 ⎠=⎝
−1
3
3
3
−5
⎟
2⎠
3
−5
1
3
− 76
− 71
2
7
3
7
5
7
− 73
(A|I), A .
5
7
− 47
1
7
⎞
⎟
⎠
A−1 , (A−1 A|A−1 I) = (I|A−1 )
,
, A I, I A−1 . ⎛
1
2
⎜
⎝2
3
X 3
3
, ⎛ ⎞
3
⎟⎜ ⎟ ⎜ ⎟
2 ⎠ ⎝ x2 ⎠ = ⎝ 4 ⎠
x3
5
4
3
⎞⎛
x1
⎞
A−1 (A|I|b) = (I|A−1 |A−1 b)
0
⎛
1
⎜
A = ⎝2
3
1
⎛
1
⎜
A=⎝ 0
0
2
⎛
1
⎜
A=⎝ 0
3
⎛
2
−1
−3
0
1
0
0
1
0
⎜
A=⎝ 0
0
1
0
2
3
3
3 1
2 0
4 0
1
0
3 1
−4 −2
−5 −3
−5 −3
4 2
7 3
0 − 67
2
0 7
3
1 7
⎞
0 3
⎟
0 4⎠
1 5
0
0
1
0
2
−1
−3
− 17
5
7
− 37
⎞
3
0 ⎟
0 −2 ⎠
1 −4
⎞
0 −1
⎟
0 2 ⎠
2
1 5
7
4
−7
1
7
3
7
6
7
2
7
⎞
⎟
⎠
D 850
3
,
7
x1 =
x2 = 67 , x3 =
2
.
7
A ,
.
, B −1
B B next , B −1
next , . −1
B −1
next = EB
E m B P j P r B next , E r ⎛ −(B −1 P ) ⎞
j 1
⎜ −(B −1 P j )2
⎜
⎜
..
⎜
.
1
⎜
ξ=
⎜
(B −1 P j )r ⎜
+1
⎜
..
⎜
⎝
.
⎟
⎟
⎟
⎟
⎟
,
⎟
⎟ ← r ⎟
⎟
⎠
(B −1 P j )r = 0
−(B −1 P j )m
(B −1 P j )r = 0, B −1
next .
B −1
next . F m , r (B −1 P j ) ,
F = (e1 , · · · , er−1 , B −1 P j , er+1 , · · · , em )
B next
B r B −1 P j
, B next = BF
,
−1
= F −1 B −1
B −1
next = (BF )
E = F −1 , .
B B 1 , B , . B 0 = I = B −1
0
,
−1
B −1
1 = E 1B 0 = E 1I = E 1
, B i , i B i , −1
B −1
= E i B −1
i
i−1 = E i E i−1 B i−2 = · · · = E i E i−1 · · · E 1
B,
B −1 = E n E n−1 · · · E 1
. ⎛
⎞
2 1 0
⎜
⎟
B = ⎝0 2 0⎠
4 0 1
D.2
0
⎛
1
0
⎜
B 0 = B −1
0 = ⎝0
0
0
851
⎞
1
⎟
0⎠
0
1
1
⎛
2
0
⎜
B1 = ⎝ 0
⎛
E1 =
2
⎛
2
⎜
B2 = ⎝ 0
⎛
4
⎜
B −1
1 P2 =⎝
⎛
1
0
1
0
0
1
0
0
1
2
−2
+ 21
− −2
2
⎟
0⎠,
0
1
0
0
⎛ ⎞←r=1
2
⎜ ⎟
B −1
P
=
P
=
0⎠
⎝
1
1
0
4
⎛ 1
⎞
0 0
2
⎜
⎟
B −1
0 1 0 ⎠
1 = ⎝
⎞
1
⎟
0⎠,
0
1
−2
⎞⎛ ⎞ ⎛ 1 ⎞
1
2
⎟
⎟⎜ ⎟ ⎜
0 ⎠⎝2⎠ = ⎝ 2 ⎠ ← r = 2
0
−2
1
⎞ ⎛
⎞
0
1 − 14 0
⎟ ⎜
⎟
1
0⎠ = ⎝ 0
0 ⎠
2
1
0
⎛
1
1
⎜
−1
B −1 = B −1
2 = E 2B 1 = ⎝ 0
0
0
1
1
1
⎞⎛
⎞
(q × p)
⎛
0
0
1
⎟ ⎜
0 ⎠=⎝
−2
0
1
⎟⎜
0 ⎠⎝
(q × q)
B 11
⎜ (p × p)
⎜
B=⎜
⎝ B 21
(q × p)
⎛
1
2
0
n A B, ⎛
⎞
A12
A11
⎜ (p × p) (p × q) ⎟
⎜
⎟
A=⎜
⎟ , A11 ⎝ A21
A22 ⎠
B A 1
0
− 14
1
2
0
⎞
⎟
0⎠ = B
1
2
⎞
1
+ 21
⎜ 0
⎝ −2
− 24
2
−2
1
⎜
E2 = ⎝ 0
0
4
0
B 12
⎞
(p × q) ⎟
⎟
⎟,
B 22 ⎠
(q × q)
, AB = I n , A11 B 11 + A12 B 21 = I p
A11 B 12 + B 12 A22 = 0
1
2
⎞
− 14
0
0
1
2
⎟
0 ⎠
−2
1
1
D 852
BA = I n , B 21 A11 + B 22 A21 = 0
B 21 A12 + B 22 A22 = I q
A11 , A−1
11 . B 11 , B 12 , B 21 , B 22 , −1
−1
B 11 = A−1
(A21 A−1
11 + (A 11 A12 )D
11 )
−1
B 12 = −(A−1
11 A 12 )D
B 21 = −D −1 (A21 A−1
11 )
B 22 = D −1
D = A22 − A21 (A−1
11 A 12 )
, ⎛
1
⎜
A = ⎝2
3
2
⎟
2⎠
3
4
A11 = (1), A12 = (2, 3), A21 =
,
2
3
, A22 =
3
2
3
4
6
B 11 = −
7
= 1, 3 2
2
−1 −4
D=
−
(1)(2, 3) =
3
−3 −5
3 4
5
−5
4
− 47
1
−1
7
D =−
=
1
7
− 37
3 −1
7
, B 12 =
1
−
7
5
7
, B 21 =
2
7
3
7
, B 22 =
5
7
− 37
− 47
1
7
B = A−1 .
D.2.8
A−1
11
⎞
3
3
Excel Excel (1) ;
D.1 ,
E4:F6 ,
(2) ;
(3) (2 × 3) ;
.
( excelMatManip.xls). 1() , A (4) A4:C5. Transpose(A), AT , D.3
D.1
(1) E4 (2)
()
Excel 853
( excel Mat Manip.xls)
=TRANSPOSE(A4:C5);
E4:F6;
(3) F2 ;
(4) CTRL+SHIFT+ENTER .
2 , A B A10:C13 A16:A18 , =MMULT(A10:C13, A16:A18), E10:E13. E10
1 2 4 ( E4:F6
E10:E13). A10:C13) . 3 , A22:C24 =MINVERSE(A22:C24) , 4 , E28 E22:G24 , MMULT(A16:A18,
E22 1 2 =MDETERM(A28:C30), .
D.3
X = (x1 , x2 , · · · , xn )T
4 . A28:C30 D 854
⎛
a11
⎜
⎜ a21
A=⎜
⎜ ..
⎝ .
an1
a12
···
a22
..
.
···
an2
···
Q(X) = X T AX =
a1n
⎞
⎟
a2n ⎟
.. ⎟
⎟
. ⎠
ann
n
n aij xi xj
i=1 j=1
. aij +aji
2
A , Q(X) .
. ⎛
1
⎛
⎞⎛
x1
⎞
3
0
2
1
1
2
2
. 1
⎟⎜ ⎟
6 ⎠ ⎝ x2 ⎠
⎜
Q(X) = (x1 , x2 , x3 ) ⎝ 1
, 2 , A 0
7
⎜
Q(X) = (x1 , x2 , x3 ) ⎝ 2
aij aji (i = j) x3
⎞⎛
x1
⎞
7
⎟⎜ ⎟
3 ⎠ ⎝ x2 ⎠
3
2
x3
A .
(1) , X = 0, Q(X) > 0.
(2) , X, Q(X) 0, X = 0,
Q(X) = 0.
(3) , −Q(X) .
(4) , −Q(X) .
(5) , .
, .
(1) Q(X) (), A (
,
(2) Q(X) ) . A ().
, A k (−1)k , k = 1, 2, · · · , n. ,
(3) Q(X) A ().
, A k 1, 2, · · · , n.
An×n k a11 a12 · · ·
a21 a22 · · ·
.
..
.
.
.
a
a
···
k1
k2
a1k a2k .. , k = 1, 2, · · · , n.
. akk , (−1)k , k =
D.4
D.4
855
f (X ) , X 1 X 2 , f (λX 1 + (1 − λ)X 2 ) < λf (X 1 ) + (1 − λ)f (X 2 )
0 < λ < 1. , f (X ) , −f (X) .
() ( D.3)
f (X ) = CX + X T AX
C , A , A . , A , f (X) , f (X) .
1.
⎛
⎞ ⎛
⎞ ⎛
⎞
1
−2
1
⎜
⎟ ⎜
⎟ ⎜
⎟
(a) ⎝ −2 ⎠ ⎝ 4 ⎠ ⎝ −2 ⎠
−2
3
2.
⎛
1
4
⎜
A=⎝ 2
9
⎛
−1
7
⎟
⎜
−8 ⎠ , B = ⎝ 9
5
3
⎞
7
⎞ ⎛
4
2
⎟
8 ⎠
4
3
⎞
2
6
10
(a) A + 7B
(c) (A + 7B)T
(b) 2A − 3B
3. 2 , AB = BA.
4. ⎛
1
⎜
⎜ 2
A=⎜
⎜ 3
⎝
4
5
−6
7
9
⎞
⎛
2
⎟
⎟
9 ⎟
⎜
,
B
=
1
⎝
2 ⎟
⎠
3
1
7
3
−4
2
6
1
0
5
⎞
⎟
7 ⎠
9
AB.
5. 2 , A−1 B −1 .
(a) (c) 6. .
.
⎛
2
⎜
B = ⎝0
4
1
2
⎞
(b) .
(d) .
⎛
2
⎜
⎟
1 ⎠ , B −1 = ⎝
0
5
⎞
⎟
⎟ ⎜
⎜
⎜ −3 ⎟ ⎜ −6 ⎟
⎟
⎟ ⎜
⎜
⎜ 4 ⎟ ⎜ 8 ⎟
⎠
⎠ ⎝
⎝
10
5
(b)
−1
⎛
2
5
4
1
2
−1
− 85
1
4
1
2
− 83
⎞
⎟
− 41 ⎠
1
2
856
D 3 P 3 V 3 = P 1 + 2P 2 7.
(a)
.
.
(b) x1 + 2x2 = 5
x1 + 2x2 = 3
−x1 − 2x2 = −5
x1 + 4x2 = 2
(c)
,
.
x1 + 7x2 + x3 = 5
4x1 + x2 + 3x3 = 8
x1 + 3x2 − 2x3 = 3
8. B.2.7 9. A=
10. .
1
G
H
B
, B
Q(x1 , x2 ) = 6x1 + 3x2 − 4x1 x2 − 2x21 − 3x22 −
27
4
11. Q(x1 , x2 , x3 ) = 2x21 + 2x22 + 3x23 + 2x1 x2 + 2x2 x3
12. f (x) = ex .
13. f (x1 , x2 , x3 ) = 5x21 + 5x22 + 4x23 + 4x1 x2 + 2x2 x3
.
14. 13 , −f (x1 , x2 , x3 ) .
Hadley, G., Matrix Algebra, Addison-Wesley, Reading, MA, 1961.
Hohn, F., Elementary Matrix Algebra, 2nd ed., Macmillan, New York, 1964.
Press, W., B. Flannery, B. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of
Scientific Computing, Cambridge University Press, Cambridge, England, 1986.