1.2-1.4 行列式的性质

IP

59.77.1.116;

§1.2-§1.4

(

10);

Cramer1

|A|=a11a22···ann(

|A| a11 0|A|= ···

0

n

a12a22···0

············aii

a1n a2n ··· ann

|A|

a11 a21

|A|= ···

an1

0a22···an2············

).

0 0 ··· ann

(1)|A|

n=1

|A|=a11

n 1

n

|A|.

|A|=a11M11.

M11

n 1

M11=a22a33···ann,

|A|=a11a22···ann.

(2)|A|

|A|=a11M11 a21M21+···+( 1)n+1an1Mn1.

n 1

Mi1,i>1,

Mi1

Mi1=0,i>1,|A|

M11

M11=a22···ann.|A|

|A|=a11a22···ann.

2

2

n=1

n 1

|A|

i

|A|=a11M11 a21M21+···+( 1)n+1an1Mn1

Mj1(j=i)

Mj1=0,

ai1=0,

ai1Mi1=0,

|A|=0.i

|A|

i=1,

|A|=0.

i>1,

|A|=a11M11 a21M21+···+( 1)n+1an1Mn1

1

Mj1

Mj1=0,

|A|=0.

2

3

|A|

c

|B|,|B|=

c|A|.

n=1

(1)

|B|

i

|A|

i

c,

|A|

|B|=a11N11 a21N21+···+( 1)i+1cai1Ni1+···+( 1)n+1an1Nn1

Nr1

|B|

r

Nr1=cMr1,r=i,Ni1=Mi1

Mr1,Mi1(2)

|A|

|B|=c|A|.

|B|

1

|A|

1

c

|B|

|B|

i

(i>1)

|A|

i

c(

2

|A|

4

).

n

a11

a12a21)= a21

a12 ,a22 —

n=2

a21 a11

n 1

a22 =a12a21 a11a22= (a11a22 a12

n

|B|

|A|

r

r+1

Nij

|B|

i

j

|B|=a11N11 a21N21+···+( 1)r+1ar+1,1Nr1+( 1)r+2ar1Nr+1,1+···+( 1)n+1an1Nn1.

i=r,r+1,

Ni1= Mi1.|A|

i

Nr1=Mr+1,1,Nr+1,1=Mr,1,j

|B|= |A|.

j>i.

i

i+1

i+2

j

j 1

i

2(j i) 1

|B|= |A|.

2

2

5

|A|

|A|=0.

|A|= |A|,

|A|=0.

2

i

6

|A|,|B|,|C|

n

j

aij,bij,cij,|A|,|B|,|C|

r

crj=arj+brj

(j=1,2,···,n)

(1)

cij=aij=bij(i=r,j=1,2,···,n),

|C|=|A|+|B|.

n

n=1

n 1

|C|=a11Q11 a21Q21+···+( 1)r+1(ar1+br1)Qr1+···+( 1)n+1an1Qn1

Qij

|C|

i=r,

Qi1

(1),

Qi1=Mi1+Ni1,

Mi1,Ni1

7

|A|,|B|

i=r,Qr1=Mr1=Nr1,

|C|=|A|+|B|.

2

c

6,

3,

5

2

|A|=0.

5′

|A|

|A|

1

|A|

|A|=0.

|A|

1

r

|A|

1

0,

|A|=0.

|A|

1

as1=0.

1

s

|A|

|A|=0

|A|

1

ai1

|A|=0.a11=0,|A|

a11···a11··· a21···a21··· ············ an1···an1···

7,|C|=|A|.

|C|

|C|=a11Q11,

Q11

n 1

2,Q11=0,n

|C|=0,∴|A|=0.

2

6|A|,|B|,|C|

′

|C|r

|A|

r

|B|r

cir=air+bir

(i=1,2,···,n).

j=r

cij=aij=bij,

|C|=|A|+|B|.1,

|C|

r=r>1,

|C|

|C|=a11Q11 a21Q21+···+( 1)n+1an1Qn1.

Qi1

Qi1=Mi1+Ni1

Qi1,Mi1,Ni1

|C|,|A|,|B|

|C|=|A|+|B|.

7

′

c6′

,

3

5

′

′

41

s

|B|,

|B|= a1s···a11···a1n a1s a11···a11···a1na2s···a21···a2n ··············· ans···an1···ann = a2s a21···a21···a2n ···············

ans an1···an1···a =

a1s a11···a1s···a1n nn a11···a1s···a1 na2s a21···a2s···a2n ··············· = a21···a2s···a2n ···············

a ns an1···ans···ann an1···ans···ann = a···a

111s···a1n a21···a2s···a2n ···············

ns···ann = |A|

an1···a

4

2

2

2

|B|

a11 a|A|

r

s

···a1r···a···a1s···a······ a1n a|C|A|,|B|C|

0.

8(

|A|= 21

2r2s2n ···

·················· an1···anr···ans···ann ···a1s···a1r···a1n

|B|= a11 a21

···a2s···a2r···a2n ·················· ···ans

···anr

···

ann ···

an1

···a1r+a1s···a1r+a1s···a1n= a11 a21

···a2r+a2s···a2r+a2s···a2n ···

·················· an1···anr+ans···anr+ans

···

ann ···a1r···a1r+a1s···a 1n

= a11 a21···a2r···a2r+a2s···a2n

···

·················· an1···anr

···anr+ans

···ann ···a1s···a1r+a1s···a 1n

+ a11 a21···a2s···a2r+a2s···a2n

···

·················· an1

···ans···anr+ans···ann ···a 1n

=|A|+

a11···a1r···a1r a21···a2r···a2r

···a2n

···············

······ a

n1···anr···anr···ann ···a 1

n+|B|+

a11···a1s···a1s a21···a2s···a2s

···a2n

···············

······ an1···ans···ans

···ann 0,|C|

r

s

|C|=|A|+|B|=0,

|A|= |B|.r

)

|A|=a1rA1r+a2rA2r+···+anrAnr

5

2

||

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

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

|

n

a11x1+a12x2+···+a1nxn=b1

a21x1+a22x2+···+a2nxn=b2

···

an1x1+an2x2+···+annxn=bn

(2)

|A|

ξ1,ξ2,···,ξn,

a11 a21

|A|= ···

an1

a12a22···an2

············

1

a1n a2n ··· ann

A11,A21,···,An1,

A11

(1)

1

A21(1)2···,An1(1)n

a11A11ξ1+a12A11ξ2+···+a1nA11ξn=b1A11

a21A21ξ1+a22A21ξ2+···+a2nA21ξn=b2A21

···

an1An1ξ1+an2An1ξ2+···+annAn1ξn=bnAn1

(3)

(a11A11+a21A21+···+an1An1)ξ1+(a12A11+a22A21+···+

b1a12

b2a22

|A|ξ1= ······

bnan2

an2An1)ξ2+···+(a1nA11+a2nA21+···+annAn1)ξn=b1A11+b2A21+···+bnAn1.

10

············j

A1.

|A|=0,

ξ1=

A1

a1n a2n ,··· ann

|A|

,2≤j≤n,

Aj

n

|A|

b1,b2,···,bn

(Cramer

)n

a11x1+a12x2+···+a1nxn=b1

a21x1+a22x2+···+a2nxn=b2

···

an1x1+an2x2+···+annxn=bn

8

|A||A|

0,

x1=

A1

,···,xn=

An

|A|

+a12A2

|A|

=

1

|A|

[a11(b1A11+b2A21+···+bnAn1)+a12(b1A12+b2A22+···+bnAn2)

+···+a1n(b1A1n+b2A2n+···+bnAnn)]=

1

b|A||A|1

=b1.

Ai

xi=

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