1.2-1.4 行列式的性质(6)

Air

|A|

r

r 1

r 1

r 2r

···,1

2

1

r 1

|B|,

a1r a2r

|B|= ···

anr

a11a21···an1a12a22···an2············

a1n a2n ··· ann

( 1)r 1|A|=|B|=a1rN11 a2rN21+···+( 1)n+1anrNn1

=a1rM1r a2rM2r+···+( 1)n+ranrMnr

|A|=( 1)1+ra1rM1r+( 1)2+ra2rM2r+···+( 1)n+ranrMnr=a1rA1r+a2rA2r+

···+anrAnr,

Ni1,Mir

|B|

|A|

2

1 0 a21

|A|= ···

an1

0a22···an2

···0···a2s 1·········ans 1

a1sa2s···ans

0a2s+1···ans+1

···0 ···a2n =a1sA1s.······ ···ann

|A|

s

|A|=a1sA1s+a2sA2s+···+ansAns,

Ais(i>1)

2

a11 a21

|A|= ···

an1

0,0.

a12a22···an2

6

′

1,

|A|=a1sA1s.2

···a1n ···a2n ,|A|=a11A11+a12A12+···+a1nA1n.······ ···ann

a11 a21

|A|= ···

an1

0a22···an2············

=a11A11+a12A12+···+a1nA1n

0 0 a2n + a21 ··· ···

ann an1

a12a22···an2············

00

a2n +···+ a21 ······

an1ann

a22···an2············

a1n a2n ··· ann 2

8

|A|=ai1Ai1+ai2Ai2+···+ainAin.

6