70
CHAPTER 7
7.1
Aug = [A eyesizeA]
Here’s an example session of how it can be employed.
A = rand3 A =
0.9501 0.4860 0.4565 0.2311 0.8913 0.0185
0.6068 0.7621 0.8214 Aug = [A eyesizeA]
Aug = 0.9501 0.4860 0.4565 1.0000 0 0
0.2311 0.8913 0.0185 0 1.0000 0 0.6068 0.7621 0.8214 0 0 1.0000
7.2 a
[A]: 3 × 2
[B]: 3 × 3
{C}: 3 × 1
[D]: 2 × 4
[E]: 3 × 3
[F]: 2 × 3
⎣G⎦: 1 × 3
b
square: [B], [E]; column: {C}, row: ⎣G⎦
c
a
12
= 5, b
23
= 6, d
32
= undefined, e
22
= 1, f
12
= 0, g
12
= 6
d
MATLAB can be used to perform the operations 1
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎣ ⎡
= +
9 5
9 3
8 13
8 5
] [
] [
B E
2 ⎥
⎥ ⎦
⎤ ⎢
⎢ ⎣
⎡ −
− −
− =
− 1
3 3
1 6
1 2
3 ]
[ ]
[ B
E
3 [A] + [F] =
undefined 4 ⎥
⎥ ⎦
⎤ ⎢
⎢ ⎣
⎡ =
20 5
30 10
5 35
15 20
] [
5 F
5 [A] × [B]
= undefined 6
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎣ ⎡
= ×
29 24
45 36
68 54
] [
] [
A B
7 [G] × [C]
= 56
8
⎣ ⎦
1 6
2 ]
[ =
T
C
9 ⎥
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎢ ⎣
⎡ =
5 6
7 3
1 4
2 5
] [
T
D 10
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎣ ⎡
= ×
4 1
6 2
1 7
3 4
] [B
I
7.3
The terms can be collected to give
71
⎪⎭ ⎪
⎬ ⎫
⎪⎩ ⎪
⎨ ⎧
− =
⎪⎭ ⎪
⎬ ⎫
⎪⎩ ⎪
⎨ ⎧
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎣ ⎡
− −
− 40
30 10
7 3
4 7
4 3
7
3 2
1
x x
x
Here is the MATLAB session: A = [-7 3 0;0 4 7;-4 3 -7];
b = [10;-30;40]; x = A\b
x = -1.0811
0.8108 -4.7490
AT = A AT =
-7 0 -4 3 4 3
0 7 -7 AI = invA
AI = -0.1892 0.0811 0.0811
-0.1081 0.1892 0.1892 0.0618 0.0347 -0.1081
7.4
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎣ ⎡
− −
= ×
24 17
56 55
8 23
] [
] [
Y X
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎣ ⎡
− −
= ×
2 23
52 30
8 12
] [
] [
Z X
⎥⎦ ⎤
⎢⎣ ⎡
− =
× 34
47 8
4 ]
[ ]
[ Z
Y
⎥⎦ ⎤
⎢⎣ ⎡
− =
× 32
20 16
6 ]
[ ]
[ Y
Z
7.5 Terms can be combined to yield
72 g
m kx
kx g
m kx
kx kx
g m
kx kx
3 3
2 2
3 2
1 1
2 1
2 2
= +
− =
− +
− =
−
Substituting the parameter values
⎪⎭ ⎪
⎬ ⎫
⎪⎩ ⎪
⎨ ⎧
= ⎪⎭
⎪ ⎬
⎫ ⎪⎩
⎪ ⎨
⎧ ⎥
⎥ ⎦
⎤ ⎢
⎢ ⎣
⎡ −
− −
− 525
. 24
43 .
29 62
. 19
10 10
10 20
10 10
20
3 2
1
x x
x
A MATLAB session can be used to obtain the solution for the displacements K=[20 -10 0;-10 20 -10;0 -10 10];
m=[2;3;2.5]; mg=m9.81;
x=K\mg x =
7.3575 12.7530
15.2055
7.6 The mass balances can be written as
5 55
54 2
25 1
15 5
54 4
44 3
34 2
24 03
03 3
34 31
2 23
2 25
24 23
1 12
01 01
3 31
1 12
15
= +
+ −
− =
− +
− −
= +
+ −
= +
+ +
− =
− +
c Q
Q c
Q c
Q c
Q c
Q c
Q c
Q c
Q c
Q Q
c Q
c Q
Q Q
c Q
c Q
c Q
c Q
Q
The parameters can be substituted and the result written in matrix form as
⎪ ⎪
⎭ ⎪⎪
⎬ ⎫
⎪ ⎪
⎩ ⎪⎪
⎨ ⎧
= ⎪
⎪ ⎭
⎪ ⎪
⎬ ⎫
⎪ ⎪
⎩ ⎪
⎪ ⎨
⎧
⎥ ⎥
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎢ ⎢
⎣ ⎡
− −
− −
− −
− −
160 50
4 1
3 2
11 8
1 9
1 3
3 1
6
5 4
3 2
1
c c
c c
c
MATLAB can then be used to solve for the concentrations Q = [6 0 -1 0 0;
-3 3 0 0 0; 0 -1 9 0 0;
0 -1 -8 11 -2; -3 -1 0 0 4];
Qc = [50;0;160;0;0];
73 c = Q\Qc
c = 11.5094
11.5094 19.0566
16.9983 11.5094
7.7 The problem can be written in matrix form as
