1.2-1.4 行列式的性质(8)

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