線性代數(shù)1 1二階與三階行列式

§1.1

二階與三階行列式補(bǔ)充例題首頁上頁返回下頁結(jié)束鈴一、二元線性方程組與二階行列式二、三階行列式提示:

[a11x1+a12x2=b1]

·a22

a11a22x1+a12a22x2=b1a22

,[a21x1+a22x2=b2]

·a12

a12a21x1+a12a22x2=a12b2

,

(a11a22-a12a21)x1=b1a22-a12b2

.一、二元線性方程組與二階行列式x

+a x

=b21

1

22

2

2

a用消元法解二元線性方程組

a11x1+a12x2=b1,得11

22

12

21下頁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、二元線性方程組與二階行列式11

22

12

2111

2

1

212a

a

-a

aa b

-b

a,

x

=.x

+a x

=b21

1

22

2

2

a用消元法解二元線性方程組

a11x1+a12x2=b1,得11

22

12

21下頁1

22

12

21a

a

-a

ab

a

-a

bx

=b22a1a11

a22

12

2a221a11b1a2

b2a

aa111

12

2a22x

=

———a

—

,x2=

—1

———

.a212a2b1

2a1我們用符號

a11

a1表示代數(shù)和a

a11

22-a12a21,這樣就有一、二元線性方程組與二階行列式11

2

1

212a

a

-a

aa b

-b

a.x

+a x

=b21

1

22

2

2

a用消元法解二元線性方程組

a11x1+a12x2=b1,得11

22

12

21

11

22

12

21下頁1

22

12

21a

a

-a

ab

a

-a

b,

x

=x

=11aa

a12

2a22112

21表示代數(shù)和a11a22-a

a

,并稱它為二階行a21我們用

a11

a12a22行列式中的相關(guān)術(shù)語行列式的元素、行、列、主對角線、副對角線.對角線法則二階行列式是主對角線上兩元素之積減去的副對角線上二元素之積所得的差.列式.=a

a

-a

a

.11

22

12

21下頁一、二元線性方程組與二階行列式1

2

2x1

+

x2

=1

3x

-2x

=12例1

求解二元線性2方程組

.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

.下頁11

22

12

21為了便于記憶和計(jì)算,我們用符號a21a11

a12

a13a22a32a23

表示代數(shù)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方程組

a x

+a x

+a x

=b

的解為

x

=D

D

D

DD

2

31

,x

=

2

,x

=3

,a11a22a33+a12a23a31+a13a21a32-a11a23a32-a12a21a33-a13a22a31.二、三階行列式下頁我們用符號a21a23

表示代數(shù)和a11

a12

a13a22a31

a32

a33=a1a11a22a33+a12a23a31+a13a21a32-a11a23a32-a12a21a33-a13a22a31,并稱它為三階行列式.行列式中的相關(guān)術(shù)語行列式的元素、行、列、主對角線、副對角線.對角線法則1a22a33+a12a23a31+a13a21a32-a11a23a32-a12a21a33-a13a22a31.二、三階行列式下頁=a1列式

D=

-2

2

1

.-3

4

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

2

-4例2

計(jì)算三階行解

按對角線法則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.下頁例3

求解方程1

1

12

3

x4

9

x2=0

.解方程左端的三階行列式D=3x2+