Principles of Mathematics in Operations Research

Solutions 231

2 1 0

1 2 1

0 1 1

= 1 = 106 det(^) > 0!

The 3x3 minor, itself, is OK as well,

iv. All the pivots (without row exchanges) satisfy di > 0;

"2 10"

12 1

01 1

<-*

"2 1 0'

0|1

0 1 1_

<-)•

"2 1 0"

0*1

L°°|J

di T|>0,d^A>o,rf3 = ^>0!

v. 3 a possibly singular matrix W B A = WTW;

A

and W = i

1

loo

"2 10"

1 2 1

01 1 -{i

"1 10"

0 1 1

001 }(i

"100"

1 10

01 1

= WTW

100

1 10

01 1

is nonsingular!

5.3

V/(x) =

1L

dxi

dx 2

x\ + Xi + 2X2

2xx + x 2 - 1

xi = 1 — 2xi

(an - l)(ari - 2) = 0

Therefore,

xA —

1

-1 , xB =

2

-3

are stationary points inside the region defined by — 4 < x 2 < 0 < xi < 3.
Moreover, we have the following boundaries

xi =

" 0 "

LX2\

, XII =

' 3 "

1^2 J

and xin = Xi -4 , Xiv

Xl

0

defined by

xc =

[ol

0 , xD =

[ ol

-4_ , xE =

[3]

0 , XF =

[ 3]

-4

Let the Hessian matrix be

V^2 f(x) =

(^2) J_
dx\dx\ dx\dx^
a^2 f
dxzdxi Qx^dx^
2xi + 1 2
2 1