Principles of Mathematics in Operations Research

226 Solutions

4.2 Let

A =

"1 1

2 1

0 1

1 -1

2 -2

-1

1

1

1

2

-1

2

0

3

2

-1

1

-1

1

4

d(s) 2)^5 , fc = 1, Ax =2, m =5.

Ai = A - 11 =

-1 1 -1 -1 -1

2-1121

0 1-1 0-1

1-1111

2-2222

=> dimN{A{) = 5 - rank^) = 5-3 = 2.

A\ = 0 => dimN{A\) = 5 => mj = 2, m(s) = (s - 2)^2.

Choose v 2 e JV(-AI) 9 Axv 2 ^ 0.

«2=e? = (l,0,0,0,0)r => v 1 =A 1 v 2 = (-l,2,0,l,2)T.

Choose v 4 ^ OT 2 3 a ^ 0, v 4 e Af(Al) 3 Axv 2 / 6.

v 4 = e\ = (0,1,0,0,0)^2 V3 AlV4 = (l,-l,l,-l,2)T.

Choose W5 € Af(Ai) independent from v\ and vz-

V5 = (1,0,0)-1,0)T.

Thus,

5 =

-11 10 1

2 0-11 0

00 10 0

10-10-1

2 0-20 0

S-lAS =

"2 1

2

2 1

2

2

4.3

A =

-A-10 lo r -i- 0 "

0 — — v

10 10

0 0 *>.

=* d(s) = (a - — J , fc =

A1 = A-l/ =

"o ^ o"

0 0i

0 0 0

= 1,A =

1

To'

n = 3

dimM(Ai) = 3 - ranfc(A 1 ) = 3 - 2 1.