Principles of Mathematics in Operations Research

224 Solutions

Problems of Chapter 4

4.1 In order to prove that

det A = an An + a^A^ H h ainAin,

(property 11) where Aij's are cofactors (Aij = (—l),+J'detMjj, where the

minor Mij is formed from A by deleting row i and column j);

without loss of generality, we may assume that i = 1.

Let us apply some row operations,

On «22 «13

«21 ^22 «23

«31 a22 «33

Onl «n2 On3

Oln

«2n

«3n

->•

Oil «22 «13 - - - 0,\n

0 022 «23 ' • • a 2 „

0 a 3 2 Q33 • • • Ctzn

0 a„2 «n3 "• «r

where atj = "aija;^1 +a^ai1, i,j = 2,..., n. In particular, a 22 = -a'*a»o»+^0 «°n.

Furthermore,

>!->•

an ^22 ^13 '

0 Q22 C*23 •

0 «22 «33 •

0 an2 an3 •

• 0,in


• a2n


• a3n

Otnn _

—•

an «22 «13 • •

0 C*22 Q23 • '

0 0 /? 33 • •

din

OLln

fan

0 0 /3„ 3 • • • Pn

where /3y = -"'J^0 "^"", M = 2,..., n. In particular,

-0:230:32 + «33Q22

/3 33 =

«22

(ai2«3i ~ fl32Qii)(a23aii - Q13Q21) + (033011 ~ Qi303i)(o220ii - 012O21)

«ii(022011 - ai 2 a 2 i)

If we open up the parentheses in the numerator, the terms without an cancel
each other, and if we factor an out and cancel with the same term in the
denominator, we will have

A»3 =

Q12Q23Q31 + Q13Q32Q21 — OllQ23032 ~ 013031022 ~ 012021033 + Q11Q22Q33

~a 12 a1l + 022&11

If we further continue the row operations to reach the upper triangular form,
we will have

A-+ •

an

0

0

0

022 «13 "

Q22 a23 •

0 /3 33 •

0 0 •

• oi„


• ain

Snn