Principles of Mathematics in Operations Research

Solutions 223

P 3 a:-6|| =

error.

[2.4201

4.3503

6.2805

[8.2107

"2"

5

6

.8.

=

0.4201

-0.6497

0.2805

0.8495

= 0.8695 is the minimum

(d)

A 4 =

-10 0 1

1-1 00

0 1-10

0 0-11

Ai =

-110 0

0-110

0 0-1-1

10 0 1

AiAt =

2-1 0-1

-1 2-1 0

0-1 2-1

-1 0-1 2

Clearly, AJA4 is not invertible. Then, we resort to the singular value decom-
position A4 = Q1SQ2 , where

Qi

I \/2

'2 2 o|

(^0) u -& i
^2 2 2
2 of
V2 1
2 2
, £ =
"4 0 0 0"
0200
0020
.0000.
I Q2
_I v^

2 2 of

0 -^ i u

2 2

(^2) of
\/2 1
2 2
Then, x = Q2^Qjbi finds the solution:
IE =
_1 1 _1 1
2 2 2 2
-^ 0^0
0-^ 0^
1 III
2 2 2 2 J
1000
0100
0 0 i 0
0 0 00
1
"2
/2
2
0
1
2
1 1 1 "
2 2 2
0 & 0
^2 n ^/2
2 U 2
1 1 1
2 2 2 J
2
4
3
3
=>
Xi
^2
X3
X4
0.22855
-0.42678
-0.25000
0.12500