线性代数1 1二阶与三阶行列式

§1.1

二阶与三阶行列式补充例题首页上页返回下页结束铃一、二元线性方程组与二阶行列式二、三阶行列式提示:

[a11x1+a12x2=b1]

·a22

a11a22x1+a12a22x2=b1a22

,[a21x1+a22x2=b2]

·a12

a12a21x1+a12a22x2=a12b2

,

(a11a22-a12a21)x1=b1a22-a12b2

.一、二元线性方程组与二阶行列式x

+a x

=b21

1

22

2

2

a用消元法解二元线性方程组

a11x1+a12x2=b1,得11

22

12

21下页1

22

12

21a

a

-a

ab

a

-a

bx

=,提示:

[a11x1+a12x2=b1]

·a21

a11a21x1+a12a21x2=b1a21

,[a21x1+a22x2=b2]

·a11

a11a21x1+a11a22x2=a11b2

,

(a11a22-a12a21)

x2=a11b2-b1a21

.一、二元线性方程组与二阶行列式11

22

12

2111

2

1

212a

a

-a

aa b

-b

a,

x

=.x

+a x

=b21

1

22

2

2

a用消元法解二元线性方程组

a11x1+a12x2=b1,得11

22

12

21下页1

22

12

21a

a

-a

ab

a

-a

bx

=b22a1a11

a22

12

2a221a11b1a2

b2a

aa111

12

2a22x

=

———a

—

,x2=

—1

———

.a212a2b1

2a1我们用符号

a11

a1表示代数和a

a11

22-a12a21,这样就有一、二元线性方程组与二阶行列式11

2

1

212a

a

-a

aa b

-b

a.x

+a x

=b21

1

22

2

2

a用消元法解二元线性方程组

a11x1+a12x2=b1,得11

22

12

21

11

22

12

21下页1

22

12

21a

a

-a

ab

a

-a

b,

x

=x

=11aa

a12

2a22112

21表示代数和a11a22-a

a

,并称它为二阶行a21我们用

a11

a12a22行列式中的相关术语行列式的元素、行、列、主对角线、副对角线.对角线法则二阶行列式是主对角线上两元素之积减去的副对角线上二元素之积所得的差.列式.=a

a

-a

a

.11

22

12

21下页一、二元线性方程组与二阶行列式1

2

2x1

+

x2

=1

3x

-2x

=12例1

求解二元线性2方程组

.解由于D

=

3

-2

=3-(-4)

=7

„

0

,2

11

11D

=12-2

=12-(-2)

=14

,2

12D

=

3 12

=3-24

=-21,因此114D

71x

=22-21D

7D

D=

=

2

,

x

== =-3

.11a1a

a12

2a2=a

a

-a

a

.下页11

22

12

21为了便于记忆和计算,我们用符号a21a11

a12

a13a22a32a23

表示代数和a31

a33其中

D=a11a22a33+a12a23a31+a13a21a32-a11a23a32-a12a21a33-a13a22a31,D1=b1a22a33+a12a23b3+a13b2a32-b1a23a32-a12b2a33-a13a22b3,D2=a11b2a33+b1a23a31+a13a21b3-a11a23b3-b1a21a33-a13b2a31,D3=a11a22b3+a12b2a31+b1a21a32-a11b2a32-a12a21b3-b1a22a31.

a11x1+a12x2+a13x3=b1x

+a x

+a x

=b

a21

1

22

2

23

3

2

31

1

32

2

33

3

3

D

1方程组

a x

+a x

+a x

=b

的解为

x

=D

D

D

DD

2

31

,x

=

2

,x

=3

,a11a22a33+a12a23a31+a13a21a32-a11a23a32-a12a21a33-a13a22a31.二、三阶行列式下页我们用符号a21a23

表示代数和a11

a12

a13a22a31

a32

a33=a1a11a22a33+a12a23a31+a13a21a32-a11a23a32-a12a21a33-a13a22a31,并称它为三阶行列式.行列式中的相关术语行列式的元素、行、列、主对角线、副对角线.对角线法则1a22a33+a12a23a31+a13a21a32-a11a23a32-a12a21a33-a13a22a31.二、三阶行列式下页=a1列式

D=

-2

2

1

.-3

4

-2,有1a22a33+a12a23a31+a13a21a32-a11a23a32-a12a21a33-a13a22a31.1

2

-4例2

计算三阶行解

按对角线法则D

=1·2·(-2)+2·1·(-3)+(-4)·(-2)·4-1·1·4

-2·(-2)·(-2)

-(-4)·2·(-3)=-4-6+32-4-8-24

=-14.下页例3

求解方程1

1

12

3

x4

9

x2=0

.解方程左端的三阶行列式D=3x2+