1.2-1.4 行列式的性质(7)

2

4,

|A|

8 a11 a12′

|A|=a21a22······ an1 an2 .2

|A|

′

|A|

···

·········a1n

a2n···

ann

9

|A′

|=|A|.

n=1

Mij,Nij

Nij

Mji

Nij=Mji,

|A′

|

|A′

|=a11N11 a21N12+···+( 1)n+1an1N1n=a11M11 a21M21+···+( 1)n+1an1Mn1=|A|.2

10

|A|

n

i

j

aij

r,1≤r≤n,

a1rA1r+a2rA2r+···+anrAnr=|A|;a1rA1s+a2rA2s+···+anrAns=0;s=r;ar1Ar1+ar2Ar2+···+arnArn=|A|;ar1As1+ar2As2+···+arnAsn=0;s=r.

|A|

r

s···a1r···a1r···a1n|B|= a11 a21

···a2r···a2r···a2n

···

·················· an1

···anr···

anr

···ann =0|B|

s

a

1rA1s+a2rA2s+···+anrAns=0.

Cramer

7

A|′

|A|

1

Aij,

2

|