Σημειωσεις Γραμμικης αλγεβρας

.

.

pi 2010, v.0.91

.

vi

I 1

1 21.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 182.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 pi 383.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 534.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 775.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 pi Gauss 906.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

i

.

ii

7 1157.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8 1418.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

9 1599.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1599.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

10 17310.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17310.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17510.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

11 19811.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19811.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 19911.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

12 21712.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21712.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21812.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

13pi 23413.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23413.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23513.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

II pi 261

14 : 26214.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26214.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26614.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

15pipi 294

16, 295

.

iii

17 296

18 297

19 29819.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29819.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 30219.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

20 320

.

pi ,

pi pi pipi pi - pi . pi pi pi . pi .

, pi pi, pi - pi pi pi pi pi pi ! pi, pi pi , pi - pi. pi pi pi . , pi. pipi pi .

pi pi pi pi pi pi . pi v.0.91 pi pi 1. pi pi , pi . pi pi pi , pi .

pi 20 . 1-13 pi pi pipi . 14-20 pi , pi pipi 2.

pi pi pi ( , -, ) pi pi . k.m.n, . n- m- pi k- . pi, pi . .

1. : .1 pi software: beta pi

v.1.00.2 pi 14-19.

iv

.

v

2. : pi pi -.

3. : pi pi - .

pi pi, pi . pi , pi .pi pi ., pi pi pi.

, pi 2010

.

( pi - , pi pi):

1. pi,

2. ,

3. N- , pi N = 1, 2, 3, ... 3.

", pi pi pi . pi - pipi. pi pi pi pi . pi pi pi - pi pi pi. pi pi pi . pi-pi, pi pi pi pi , pi pi N pipi (2 ) (3 ). pi pi pi pi pipi . pipi, , pi ,pi pi4.

pi pi 20 . 1 13 pi pi pipi . - 14 20 pi , pi pipi 5.

pi pi , pi . .

3 N = 0 pi pi pipi.4 pipi pi N = 2 3 (. pipi

) pi , -. pi pi , pi (pi.. pi).

5 14-19 ( pi v.091 ).

vi

.

vii

1. pi R C.

2.

Nn=1

an = a1 + a2 + ...+ aN .

,

Nn=1

an = a1 a2 ... aN .

3. . .. .

.

I

1

.

1

(pi ) pi - ( ). pi pi - pi pi -pi pi (pi pi -) pi pi pi.

1.1

1.1.1. pi A :

A =

a11 a12 ... a1Na21 a22 ... a2N... ... ... ...aM1 aM2 ... aMN

. A m- n- amn (A)mn( m,n).

1.1.2. , MN pi pi {1, 2, ...,M} {1, 2, ..., N} pi R:

A : {1, 2, ...,M} {1, 2, ..., N} R.. (m,n) {1, 2, ...,M} {1, 2, ..., N} amn, pi pi m,n.

1.1.3. pipi pi pi . .. pi pi 1.

1.1.4. amn pi , m n .

1.1.5. A M N , M N .1 11.

2

.

1. 3

1.1.6. A N N (. ) .

1.1.7. M N pi A,B ( A = B) m {1, ...,M} n {1, ..., N} mmn = bmn.1.1.8. A 1 ( M 1) pi-:

A =

a11a21...aM1

.1.1.9. A 1 ( 1N ) pi-:

A =[a11 a12 ... a1N

].

1.1.10. pipi- .

1.1.11. pi pi :

A =

r1r2...

rM

,pi ( m {1, ...,M} ):

rm =[am1 am2 ... amN

].

1.1.12. pi pi pi :

A =[

c1 c2 ... cN],

pi ( n {1, ..., N} ):

cn=

a1na2n...

aMn

.1.1.13. pi pi . pi A (M N ) B ( M N ).

(A + B)mn = amn+bmn (AB)mn = amnbmn.

! pi A + B A B !

.

1. 4

1.1.14. pi A MN , m = 1, 2, ...,M ,n = 1, 2, ..., N amn = 0, .

0M,N=

0 0 ... 00 0 ... ...... ... ... ...0 ... ... 0

. pi 0M,N pi 0 (. MNpipi pi , pipi )..

1.1.15. pipi pi pi :

( A)mn = amn.1.1.16. pi A,B ( pi ).

1. A + 0 = A. ( 0 pi).

2. A + B = B + A. (.)

3. A + (B + C) = (A + B) + C. (.)

4. A + ((1) A) = ((1) A) + A = 0. ( A A = (1) A.)5. (A + B) = A+ B = (A + B) . (pi.)6. (+ ) A = A+ A.(pi.)7. ( ) A = ( A) .

1.1.17. pipi pi pi pi . piA ( M K) B ( K N ).

(A B)mn =Kk=1

amkbkn.

! pipi A B A B!

1.1.18. pi A N N , m,n =1, 2, ..., N amm = 1 amn = 0 m 6= n, .

IN=

1 0 0 ... 00 1 0 ... ...0 0 1 ... ...... ... ... ... ...0 ... ... ... 1

. pi IN pi I (. Npipi pi , pipi ).

.

1. 5

1.1.19. pi A,B,C ( pi ).

1. I A = A I = A. ( I pipi pi).2. pi pi A,B pi A B 6= B A.3. A (B C) = (A B) C. (.)4. A (B + C) = A B + A C. (pi.)5. (B + C) A = B A + C A. (pi.)6. (A B) = ( A) B = A ( B) .7. A 0 = 0 A = 0.

1.1.20. NN () pi A, pi :A2 = A A, A3 = A A A ... A0 = I. : AmAn = Am+n, (Am)n = Amn. pi pi , (pi.. A1/2, A ...2).

1.2

1.2.1. pi pi ; ;

A =

[3 24 1

], B =

[3 0

], C =

[3 3 44 1 2

], D = [3] , E =

364

pi. A 2 2, B 1 2 C 2 3 D

1 1, B 3 1.1.2.2. pi

A =

1 5 24 3 68 7 0

. a11; a23; a31; 5; 8;

pi. a11 = 1, a23 = 6, a31 = 8. 5 1 2 8 3 1.1.2.3. pi

A =

[x 2y 1

], B =

[3 z4 1

] . x, y, z;

pi. A = B i, j : aij = bij. x = 3, y = 4 z = 2.2 pi pi ;;.

.

1. 6

1.2.4. pi pi .

pi. pi : , . . pi ( pipi 3- ) - pi ! pi pi , .pi pi pi , pi , . .

1.2.1

pi (, ) pi pi pi pipi. .. , api pi (x, y, z). pi pi pi- (pi a ) pi (0, 0, 0), pi pi a , pi (x, y, z). a = (x, y, z) 1-pi-1 . pi pi pi (0, 0, 0) pi (x, y, z) pi (x, y, z).

pi. a = (x, y, z) pi

a =

a1a2a3

a1 = x, a2 = y, a3 = z, a pi a . pi a pi [ a1 a2 a3 ]. , - pi pipi-pipi pi pi pi ( pi a ). pi pi -pi a pipi pi : , , pipi-.3

1.2.5. pi

A =

[4 2 32 1 0

] . pi

B =

4 5 2 36 2 4 12 0 1 0

.3 a pipipi pi .

.

1. 7

pi.

r1 =[

4 2 3], r2 =

[ 2 1 0 ]

A =

[r1r2

].

c1 =

[42

], c2 =

[21

], c3 =

[30

] A = [c1 c2 c3].

1.2.6. pi

A =

[1 24 3

], B =

[1 23 1

], C =

[2 1 33 1 2

].

pi A + B, AB,A + C.pi.

A + B =

[1 24 3

]+

[1 23 1

]=

[1 + 1 2 + 24 + 3 3 1

]=

[2 47 2

].

AB =[

1 24 3

][

1 23 1

]=

[1 1 2 24 3 3 + 1

]=

[0 01 4

].

pi A + C .

1.2.7. pi A + 0 = A.pi. m,n

(A + 0)mn = (A)mn + (0)mn = amn + 0 = amn = (A)mn .

m,n, A + 0 = A.

1.2.8. pi A + B = B + A.pi. m,n

(A + B)mn = (A)mn + (B)mn = amn + bmn = bmn + amn = (B + A)mn .

m,n, A + B = A + B.

1.2.9. pi (A + B) = A+ B = (A + B) .pi. m,n

( (A + B))mn = (A + B)mn = ((A)mn + (B)mn)= amn + bmn = ( A)mn + ( B)mn .

m,n, (A + B) = A + B.

.

1. 8

1.2.10. pi

A =

[1 24 3

], B =

[1 23 1

], C =

[2 1 33 1 2

].

pi AB, BA, AC, CA.pi.

AB =

[1 24 3

][

1 23 1

]=

[1 1 + 2 3 1 2 + 2 (1)4 1 + 3 3 4 2 + 3 (1)

]=

[7 013 5

].

BA =

[1 23 1

][

1 24 3

]=

[9 81 3

], AB.

AC =

[1 24 3

][

2 1 33 1 2

]=

[1 2 + 2 3 1 (1) + 2 1 1 3 + 2 24 2 + 3 3 4 (1) + 3 1 4 3 + 3 2

]=

[8 1 717 1 18

] pipi CA pi .

1.2.11. pi I A = A I = Api. m,n (I A)mn = (A)mn.

(I A)mn =Nk=1

imkakn = immamn = 1 amn = amn = (A)mn .

pi imk = 0 m 6= k. m,n (A I)mn = (A)mn.

1.2.12. pi A,B pi A B 6= B A.pi. , pi.., pi A,B pi 1.2.10.

1.2.13. pi A,B pi A B = B A.pi. .. pi

A =

[1 00 2

], B =

[3 00 4

],

AB =

[3 00 8

]= BA.

pi . .., pi

C =

[1 43 2

], D =

[2 00 2

],

CD =

[2 86 4

]= DC.

pi pi E,F .. EF = FE;

.

1. 9

1.2.14.

(AB) (A + B) = A2B2.pi. (AB) (A + B) = A2 + ABBAB2. pi

(AB) (A + B) = A2B2 pipi ABBA = 0, . AB = BA,. pi .1.2.15. pi A (B C) = (A B) C

pi. A,B,C KL, LM ,MN ., k, n:

(A (B C))kn =Ll=1

Akl(B C)ln =Ll=1

Akl(

Mm=1

BlmCmn)

=Ll=1

Mm=1

AklBlmCmn,

((A B) C)kn =Mm=1

(A B)km Cmn =Mm=1

(Ll=1

AklBlm)Cmn =

Mm=1

Ll=1

AklBlmCmn.

Ll=1

Mm=1

AklBlmCmn =Mm=1

Ll=1

AklBlmCmn,

. pi pi ( pi pi pi m pi l , , pi pi l pi m. , pipi, pi .1.2.16. N N pi A,B. (A,B) pi [A,B] = ABBA.

[A,B] = [B,A] [A, [B,C]] + B, [C,A] + [C, [A,B]] = 0.pi.

[A,B] = ABBA = (ABBA) = [B,A] .pi

[A, [B,C]] = [A, [BCCB]] = [A, [BC]][A, [CB]] = ABCBCAACB + CBA. pi

[B, [C,A]] = BCACABBAC + ACB,[C, [A,B]] = CABABCCBA + BAC.

pi[A, [B,C]] + B, [C,A] + [C, [A,B]] =

ABCBCAACB + CBA+BCACABBAC + ACB+CABABCCBA + BAC= 0.

.

1. 10

1.2.17.

5x 3y + z = 22x+ y 3z = 1

7x+ z = 8

pi.pi. pi

5 3 12 1 37 0 1

xyz

= 5x 3y + z2x+ y 3z

7x+ z

= 21

8

. pi

A =

5 3 12 1 37 0 1

, u = xyz

, b = 21

8

, Au = b, pi u pi A,b .1.2.18. pi . ( kgr) pi :

0.500 0.100 0.050 0.000 0.000 0.500 0.150 0.050 0.000 0.050pi 0.800 0.200 0.100 0.050 0.050

.

kgr ( Euro)

3 5 10 20 25 .

pi pipi pi pi.

pi. pi pi pi ( pi):

: 0.500 3 + 0.100 5 + 0.050 10 + 0.000 20 + 0.000 25 = 2.500, : 0.500 3 + 0.150 5 + 0.050 10 + 0.000 20 + 0.050 25 = 4.000,

pi : 0.800 3 + 0.200 5 + 0.100 10 + 0.050 20 + 0.050 25 = 6.650. pi pipi pi. , pi

A =

0.500 0.100 0.050 0.000 0.0000.500 0.150 0.050 0.000 0.0500.800 0.200 0.100 0.050 0.050

, b =

35

102025

,

.

1. 11

pipi pi

Ab =

0.500 0.100 0.050 0.000 0.0000.500 0.150 0.050 0.000 0.0500.800 0.200 0.100 0.050 0.050

35

102025

= 2. 504.00

6. 65

pi .

1.2.19. pi pi pi - . pi pi pi pi pi. .. 5261 pi pi - 5 2, 6 1. 4, . pi pi . pi pipi pi 3 pi 1 3. 3 pi 5 2.

1.2.2

pi. pi pi pi pi . () pi, ().

1.2.3

pi pi pi pi - pi

A =

0 1 1 0 0 1 11 0 0 0 1 1 01 0 1 1 0 0 00 0 1 0 0 0 10 1 0 0 1 0 01 1 0 0 0 0 01 0 0 1 0 0 1

.

A 0 1. amn = 1 m n pipi , amn = 0 pipi. pi A , pi ;

pi pi ( 1.2.2, 1.2.3 pi A) pi- . pi A pi 1 A. - , pi.., 2; pi

.

1. 12

A:

[0 1 1 0 0 1 1

]

0010001

= 2.

pi pi, 2, 2 pi 1 4. , pi . m- pi n- A. (m,n) A2:(

A2)mn

=7

k=1

amkakn.

pi x ( ). pipi pi x amkakn 1, pi amkakn = 1, pi amk = akn = 1 m- pi k- k- pi n- . , pi 2 pi m- n-. (A2)mn =

7k=1 amkakn 2 pi

m- n-. pi, (m,n) (A2)mn pi 2 pi m- n-. pi :

A2 =

0 1 1 0 0 1 11 0 0 0 1 1 01 0 1 1 0 0 00 0 1 0 0 0 10 1 0 0 1 0 01 1 0 0 0 0 01 0 0 1 0 0 1

2

=

4 1 1 2 1 1 11 3 1 0 1 1 11 1 3 1 0 1 22 0 1 2 0 0 11 1 0 0 2 1 01 1 1 0 1 2 11 1 2 1 0 1 3

.

pi pi Aj, j = 1, 2, 3, ... . ..

A3 =

0 1 1 0 0 1 11 0 0 0 1 1 01 0 1 1 0 0 00 0 1 0 0 0 10 1 0 0 1 0 01 1 0 0 0 0 01 0 0 1 0 0 1

3

=

4 6 7 2 2 5 76 3 2 2 4 4 27 2 5 5 1 2 42 2 5 2 0 2 52 4 1 0 3 2 15 4 2 2 2 2 27 2 4 5 1 2 5

pi 3 pi 1 3 7 pi 5 2 4. pi pi pi.

.

1. 13

1.2.20. a11 a12 ... a1Na21 a22 ... a2N... ... ... ...aM1 ... ... aMN

12...N

= 1

a11a21...aM1

+ 2

a12a22...aM2

+ ...+ Na1Na2N...aNN

(1.1)

[a1 a2 ... aN ]

12...N

= 1a1 + 2a2 + ...+ NaN . (1.2)pi. a11 a12 ... a1Na21 a22 ... a2N... ... ... ...aM1 ... ... aMN

12...N

=

1a11 + 2a12 + ...+ Na1N1a21 + 2a22 + ...+ Na2N

...1aM1 + 2aM2 + ...+ NaMN

= 1

a11a21...aM1

+ 2

a12a22...aM2

+ ...+ Na1Na2N...aNN

pi (1.1). (1.2) pi (1.1).

1.2.21. 1 0 ... 00 2 ... 0... ... ... ...... ... ... N

a11 a12 ... a1Na21 a22 ... a2N... ... ... ...aN1 ... ... aNN

=

1a11 1a12 ... 1a1N2a21 2a22 ... 2a2N... ... ... ...

NaN1 ... ... NaNN

. (1.3)pi. m,n:

1 0 ... 00 2 ... 0... ... ... ...... ... ... N

a11 a12 ... a1Na21 a22 ... a2N... ... ... ...aN1 ... ... aNN

mn

=[

0 ... 0 m 0 ... 0]

a1n...

am1,namnam+1,n...amN

= mamn

pi (1.3).

.

1. 14

1.3

1.3.1. pi pi pi.[1 20 1

],

[1 2 10 1 0

],

[1 2

],

[10

],

[1 + i 2 1

3 i].

pi. 2 2, 2 3, 1 2, 2 1, 2 2.)1.3.2. pi

A =

[5 8 12 3 1

] .

1.3.3. pi

A =

[x y4 1

], B =

[3 9z u

] . x, y, z, u;

pi. x = 3, y = 9, z = 4, u = 1.

1.3.4. pi A + B AB pi

A =

[1 3 12 1 0

], B =

[ 1 4 21 0 0

].

pi.[

0 7 33 1 0

],[

2 1 11 1 0

].

1.3.5. pi A + B AB pi

A =

[4 02 1

], B =

[1 4 22 3 6

].

pi. pi .

1.3.6. pi

A =

[1 45 7

], B =

[1 34 2

], C =

1 0 22 3 03 2 1

, D = 6 0 71 1 23 2 4

.pi AB, BA, CD, DC.

pi.[

17 1133 29

],[

16 2514 30

],

0 4 159 3 817 0 29

, 27 14 195 7 4

13 2 2

.

.

1. 15

1.3.7. pi

A =

[2 00 2

], B =

[1 54 2

], C =

4 0 00 3 00 0 2

, D = 3 2 47 4 5

3 2 1

.pi AB, BA, CD, DC. pi ;

pi.[

2 108 4

],[

2 108 4

],

12 8 1621 12 156 4 2

, 12 6 828 12 10

12 6 2

.1.3.8. pi

A =

4 0 00 3 00 0 2

, B = 6 3 41 1 8

8 0 4

.pi AB, BA. pi ;

pi.

24 12 163 3 2416 0 8

, 24 9 84 3 16

32 0 8

.1.3.9. pi ( pi pi )

A + (B + C) = (A + B) +C.

1.3.10. pi

(+ ) A = A+ A., ( ) A = ( A) .

1.3.11. pi ( pi pi )

(A B) = ( A) B.

1.3.12. pi A 0 = 0 A = 0.1.3.13. pi (A + I) (A + I) = A2+2A + I.1.3.14. pi , , (A + I) (B + I) = (A + I) (B + I). ;

1.3.15. pi , , (A + B) (A + B) 6= A2+2AB + B2.1.3.16.

A() =

[cos sin sin cos

] A()A() = A( + ).

.

1. 16

1.3.17. pi

A =

0 1 01 0 00 0 1

.

B =

b11 b12 b13b21 b22 b23b31 b32 b33

AB =

b21 b22 b23b11 b12 b13b31 b32 b33

, BA = b12 b11 b13b22 b21 b23b32 b31 b33

.pi pipi .

1.3.18. N N pi A pi pipi pi N N piI, i j. pi N N pi B. AB B i j pi BA B i j.

1.3.19. pi A

A2 =

[1 10 1

]

pi. pi pi : A =[

1 1/20 1

] A =

[ 1 1/20 1

].

1.3.20. N N pi A.

tr (A) =Nn=1

ann.

B M N C N M , tr (BC) = tr (CB).1.3.21. A,B N N pi, ABBA = I .1.3.22.

x 3y + z + 2u = 2x+ y 3u = 17x+ z + u = 0

pi.pi. 1 3 1 21 1 0 3

7 0 1 1

xyzu

= 21

0

.

1. 17

1.3.23. pi pi pi pi 1 3 4.

1.2.2

1.3.24. pi pi 1.2.17, 1.2.18, 1.2.19;

.

2

pi pi . pi pi pi pi ( pi ). pi pi, pi ,pi .

pi pi aa1 = 1, pi AA1= I, pi A1 N Npi A. pi a, pipi a 6= 0., pi A , pipi pipi , pi ( pi) a 6= 0.

pi pi pi pi, - pi ! pi pi .

2.1

2.1.1. NN pi A. piNN pi B BA = AB = I, B .

2.1.2. pi B pi BA = AB = I ( pi !) A A1 = B.

2.1.3. pi pi pi . .. pi . pi A1, A pipi A .

2.1.4. pi A,B pi A1, B1. .

1. (A1)1= A (. A1 A).

2. (A B)1 = B1 A1.

18

.

2. 19

2.1.5. 1 1 pi A = [a11] A1=[a111] a11 6= 0

A a11 = 0.2.1.6. 2 2 pi

A =

[a11 a12a21 a22

]

A1 =1

a11a22 a12a21

[a22 a12a21 a11

](2.1)

a12a21 a11a22 6= 0 A a12a21 a11a22 = 0.2.1.7. A N N , N {3, 4, ...}, A1 pi pi , pi pi pi (2.1) pi pi pi 4.2.1.8. NN pi A A2 pi A A ( A).2.1.9. N N pi A A3 pi A A A pi

A3 = A A A = (A2) A = A (A2) .2.1.10. , N N pi A n {1, 2, ...} n- A:

An = A A .. An

.

,A0= I.

2.1.11. pi pi pi - . . N N pi A

m,n {0, 1, 2, ...} : Am An = Am+n, (Am)n = Amn.2.1.12. pi A1 pi A: n {1, 2, ...}

An = A1A1.. A1n

.

2.1.13. pi pi pi . . N N pi A :

m,n {0,1,2, ...} : Am An = Am+n, (Am)n = Amn( pi A pipi ).2.1.14. N N pi A A2= A.2.1.15. pi Ak= A k {1, 2, 3...}.2.1.16. NN pi A pi k {1, 2, 3...} Ak= 0. k A.2.1.17. N N pi A A2= I.

.

2. 20

2.2

2.2.1. pi

A =

[1 21 3

]

A1 =[

352

515

15

].

pi. [1 21 3

][

352

515

15

]=

[13+21

512+21

513+31

5(1)(2)+31

5

]=

[1 00 1

][

352

515

15

][

1 21 3

]=

[31+(2)(1)

53223

51111

512+13

5

]=

[1 00 1

].

2.2.2. pi

A =

2 2 13 4 51 1 2

A1 =

37 57 217

371

174

72

.pi. 2 2 13 4 51 1 2

37 57 217

371

174

72

= 37 57 21

7371

174

72

2 2 13 4 51 1 2

= 1 0 00 1 0

0 0 1

.2.2.3. pi

A =

[3 51 2

].

pi.

A1 =[x11 x12x21 x22

].

pi [3 51 2

] [x11 x12x21 x22

]=

[1 00 1

].

pipi pipi pi

3x11 5x21 = 1x11 + 2x21 = 0

.

2. 21

3x12 5x22 = 0x12 + 2x22 = 1.

pi ( ) x11 = 2 x21 = 1 x22 = 3, x12 = 5.

A1 =[

2 51 3

].

pi pipi[2 51 3

] [3 51 2

]=

[1 00 1

].

2.2.4. pi

A =

1 0 21 1 11 2 0

.pi.

A1 =

x11 x12 x13x21 x22 x23x31 x32 x33

.pi 1 0 21 1 1

1 2 0

x11 x12 x13x21 x22 x23x31 x32 x33

= 1 0 00 1 0

0 0 1

. pipi pipi pi

x11 + 2x31 = 1x21 x11 + x31 = 0x11 + 2x21 = 0

x12 + 2x32 = 0

x22 x12 + x32 = 1x12 + 2x22 = 0

x13 + 2x33 = 0

x23 x13 + x33 = 0x13 + 2x23 = 1.

.

2. 22

, pi .pi pi . pi pipi

x11 =1

4, x21 = 1

8, x31 =

3

8

x12 = 12, x22 =

1

4, x32 =

1

4

x13 =1

4, x23 =

3

8, x33 = 1

8

pi

A1 =

14 12 1418

14

38

38

141

8

.pi pi pi 33 pi . pi pipi. 4 pi pi pi . pi, pi pi pipi pi .

2.2.5.

A =

0 1 00 0 11 0 0

, B = 0 0 11 0 0

0 1 0

. A B. pi ; (pi : pi pi).

pi.

AB =

0 1 00 0 11 0 0

0 0 11 0 00 1 0

= 1 0 00 1 0

0 0 1

,BA =

0 0 11 0 00 1 0

0 1 00 0 11 0 0

= 1 0 00 1 0

0 0 1

. pi A1= B. ;

pi C pipi

CA =

c11 c12 c13c21 c22 c23c31 c32 c33

0 1 00 0 11 0 0

= c13 c11 c12c23 c21 c22c33 c31 c32

. pipi C pi A C: pi , pi. pi pi D pipi

DB =

d11 d12 d13d21 d22 d23d31 d32 d33

0 0 11 0 00 1 0

= d12 d13 d11d22 d23 d21d32 d33 d31

.

.

2. 23

pipi D pi B D: pi , pi .

pipi ;

CAB =

c11 c12 c13c21 c22 c23c31 c32 c33

0 1 00 0 11 0 0

0 0 11 0 00 1 0

= c11 c12 c13c21 c22 c23c31 c32 c33

. pi C. pi,

CAB = CAA1= CI = C.

: pi C ( pi pipi) pi pi, . pi . . pi B pi pi pi A.

pi A,B pi ( . 14). pi pi pipipi pi pi, pi-pi, pi.

pi pi pipi pi ; A,B.

2.2.6.

A =

0 1 01 0 00 0 1

. pi ;

pi.

AA =

0 1 01 0 00 0 1

0 1 01 0 00 0 1

= 1 0 00 1 0

0 0 1

. A1 = A ( A ). , A pi 33 pi pi pi pi.

2.2.7. pi

A =

0 1 0 00 0 1 00 0 0 11 0 0 0

.

.

2. 24

pi. A pipi pi pi () . pi pipi pi pi - . . pi

A1 =

0 0 0 11 0 0 00 1 0 00 0 1 0

.

0 0 0 11 0 0 00 1 0 00 0 1 0

0 1 0 00 0 1 00 0 0 11 0 0 0

=

0 1 0 00 0 1 00 0 0 11 0 0 0

0 0 0 11 0 0 00 1 0 00 0 1 0

=

1 0 0 00 1 0 00 0 1 00 0 0 1

.2.2.8. pi

A =

[a11 a12a21 a22

].

pi.

A1 =[x11 x12x21 x22

].

pi [a11 a12a21 a22

] [x11 x12x21 x22

]=

[1 00 1

].

pipi pipi pi

a11x11 + a12x21 = 1

a21x11 + a22x21 = 0

a11x12 + a12x22 = 0

a21x12 + a22x22 = 1.

pi . pi pi

x11 =1 a12x21

a11

pi pi

a21x11 + a22x21 = 0 a211 a12x21a11

+ a22x21 = 0

x21 = a21a11a22 a12a21 x11 =

a22a11a22 a12a21 .

.

2. 25

pi pi

x22 =a11

a11a22 a12a21 x12 = a12

a11a22 a12a21pi

a22a11a22 a12a21

[a22 a12a21 a11

] [a11 a12a21 a22

]=

[1 00 1

].

pi pi pi 2 2 , pi :

A1 =1

a11a22 a12a21

[a22 a12a21 a11

]. (2.2)

pipi pi pi a11a22a12a21 6= 0. a11a22a12a21 = 0, A pi ( pi ).

2.2.9. ( N ) N N pi .pi. pi [

0 00 0

] [x yz u

]=

[1 00 1

] pi

0x+ 0z = 1

0y + 0u = 0

0x+ 0z = 0

0y + 0u = 1

, pi, pi , . pi pi.

2.2.10.

A =

[1 21 2

] .

pi. pi [1 21 2

] [x yz u

]=

[1 00 1

] pi

x+ 2z = 1

y + 2u = 0

x+ 2z = 0

y + 2u = 1

, pi, pi ( ) , . pi A.

.

2. 26

2.2.11.

A =

[1 22 4

] .

pi. pi [1 22 4

] [x yz u

]=

[1 00 1

] pi

x+ 2z = 1

y + 2u = 0

2x+ 4z = 0

2y + 4u = 1

pi pipi pi 2 2x+4z = 2 ( ) , . pi A.

2.2.12. 2 2 piA =

[a11 a12a21 a22

].

pi A1 pi a11a22 a12a21 6= 0.pi. , |A| 6= 0 pi A1 pi

pi (2.2). pi

a11a22 a12a21 = 0 (2.3)

A1=[x yz u

].

pi pipi a11 = 0. pi (2.3) pi a12a21 = 0. a12 = 0, [

1 00 1

]= A1A =

[x yz u

] [0 a120 a22

] 0x+ 0y = 1

pi pi pi a21 . pi a11 .

pi pi a11a22 a12a21 = 0 a11 6= 0. pi 2.2.8,

x11 =1 a12x21

a11

(a11a22 a12a21)x21 = a21 a21 = 0 a11a22 = 0 a22 = 0.

.

2. 27

[1 00 1

]= AA1 =

[a11 a120 0

] [x yz u

] 0x+ 0z = 1

pi pi pi. pi, pi pi A1 a11a22 a12a21 = 0.2.2.13. N N pi A. pi pi N N pi B BA = AB = I, B .

pi. pi A pi , B C. pipi

AB= BA = I

AC= CA = I

pi pi pi C pi CAB = CBA = C. pi CA = I, IB = C . B = C. , pi , .

2.2.14. pi (A1)1= A.pi. B = (A1)1. pipi

BA1 = A1B = I.

pi pi pi pipi B = A. pi A1 ( pi) , (A1)1 = B = A.

2.2.15. pi (A B)1 = B1A1.pi. pi (

B1A1) A B= B1A1A B = B1I B = B1B = I(A B) (B1A1) = A B B1A1= A1I A = A1A = I.

2.2.16. pi A B AB = BA.

A1B1 = B1A1, BA1 = A1B, AB1 = B1A.

pi.

AB = BA (AB)1 = (BA)1 B1A1 = A1B1

pi pi .

B1A1 = A1B1 BB1A1B = BA1B1B A1B = BA1

pi . pi pi.

.

2. 28

2.2.17. N N pi A. pi m,n {0, 1, 2, ...} : Am An = Am+n, (Am)n = Amn

pi. pi

Am An = A ... Am

A ... An

= A ... Am+n

= Am+n.

(Am)n =

(A ... Am

) ...

(A ... Am

)n

= A ... Amn

= Amn.

2.2.18. pi A2 , A3

A =

[1 12 1

].

pi.

A2= A A =[

1 12 1

] [1 12 1

]=

[3 00 3

],

A2= A2A =[

3 00 3

] [1 12 1

]=

[3 36 3

].

2.2.19. pi B2 , B3

B =

1 1 32 0 00 2 1

.pi. pi

B2 = B B = 1 7 62 2 64 2 1

, B3= B B B = 15 11 32 14 128 2 11

.2.2.20. pi AK K

A =

[0 11 0

]pi. pi pi pipi :[

0 11 0

]1=

[0 11 0

],

[0 11 0

]2=

[ 1 00 1

],[

0 11 0

]3=

[0 11 0

],

[0 11 0

]4=

[1 00 1

].

.

2. 29

K pi K = 4n + k, pi k = 0, 1, 2, 3. AK = A4n+k = Ak

A4n+k =

[1 00 1

] k = 0

=

[0 11 0

] k = 1

=

[ 1 00 1

] k = 2

=

[0 11 0

] k = 3.

i =1;

2.2.21.

A =

2 3 51 4 51 3 4

.

pi.

A2 = A A = 2 3 51 4 5

1 3 4

2 3 51 4 51 3 4

= 2 3 51 4 5

1 3 4

= A.:

2.2.22. pi pi

A =

2 2 41 3 41 2 3

, B = 1 1 35 2 62 1 3

, C = 0 1 14 3 4

3 3 4

, , pi pi ;

pi.

A2=

2 2 41 3 41 2 3

2 2 41 3 41 2 3

= 2 2 41 3 4

1 2 3

= A A .

pi

B2=

1 1 35 2 62 1 3

1 1 35 2 62 1 3

= 0 0 03 3 91 1 3

B3=

1 1 35 2 62 1 3

0 0 03 3 91 1 3

= 0 0 00 0 0

0 0 0

= 0

.

2. 30

B 3.

C =

0 1 14 3 43 3 4

0 1 14 3 43 3 4

= 1 0 00 1 0

0 0 1

= I C .

2.2.23. pi pi Ak= A k {1, 2, 3...}.pi. A2 = A.

A3 = A2 A = A A = A2 = A,A4 = A3 A = A A = A2 = A ...

2.2.24. pi A,B AB = BA = 0 pi A + B .

pi.

(A + B)2 = (A + B) (A + B) = A2 + AB + BA + B2 = 0

pi .

2.2.25. : AB = A BA = B, A,B .pi.

A2 = A A = AB A = A BA = A B = A. B2= B.

2.2.26. x, y

A =

1 x y1 3 51 3 5

.

pi.

A2 =

1 x y1 3 51 3 5

1 x y1 3 51 3 5

= x y + 1 3y 4x 4y 5x1 x 6 y 10

1 6 x 10 y

pi A pipi A2 = A, .

x y + 1 = 13y 4x = x4y 5x = yx 6 = 3y 10 = 56 x = 3

10 y = 5,

.

2. 31

x = 3, y = 5. pi pi pi pi. pi

A =

1 3 51 3 51 3 5

.

2.2.27. x, y

A =

4 x y1 0 14 4 3

.

pi. pipi

A2 =

4 x y1 0 14 4 3

2 = 16 4y x 4x 4y y x0 4 x 3 y

0 12 4x 13 4y

= 1 0 00 1 0

0 0 1

.pi

16 4y x = 14x 4y = 0y x = 04 x = 13 y = 0

12 4x = 013 4y = 1

pi x = 3, y = 3. pi

A =

4 3 31 0 14 4 3

.

2.2.28. : A (IA) (I + A) = 0.pi. A , . A2 = I.

(IA) (I + A) = IA + AA2 = IA2 = 0. (IA) (I + A) = 0,

IA2 = 0 A2 = Ipi A .

.

2. 32

2.2.29. A , 12

(I + A) 12

(IA) -.

pi. (1

2(I + A)

)2=

1

4

(I+2A + A2

)=

1

4(I+2A + I) =

1

4(2I+2A) =

1

2(I + A) .

pipi 12

(IA) pi pi.2.2.30. pi A

A3 =

[8 570 27

]pi. A pi 2 2.

A =

[x y0 u

].

A3 =

[x y0 u

]3=

[x3 x (uy + xy) + u2y0 u3

]pi pipi

x3 = 8

u3 = 27

x (uy + xy) + u2y = 57.

pi x = 2, u = 3, y = 3 (pi , ). pi pi pi pi

A =

[2 30 3

].

2.2.31. pi pi, pi.. A1/2 (. pi A) pi B pi B2 = A. pi pi. , Am/n pipi pi, pi pi pi, pi pi pi pi.

2.2.32. pi

A =

[1 04 1

].

pi

B =

[1 04 1

]1/2,

.

2. 33

. pi B pi pi

B2 =

[1 04 1

].

pi.

B =

[x yz u

].

B2=

[x yz u

]2=

[x2 + yz uy + xyuz + xz u2 + yz

].

pipi

x2 + yz = 1

(u+ x) y = 0

(u+ x) z = 4

u2 + yz = 1.

pi u + x = 0 y = 0. u + x = 0 pi pi . y = 0 pipi

x2 = 1

(u+ x) z = 4

u2 = 1.

pi pi x = 1 u = 1. pi .

u = 1, x = 1, z = 2, u = 1, x = 1, z = 2 pi

B1=

[ 1 02 1

], B2=

[1 02 1

], , [

1 04 1

]1/2=

[1 02 1

].

2.2.33. pi

A =

[9 48 9

].

pi

B =

[9 48 9

]1/2.

pi.

B =

[x yz u

], B2=

[x2 + yz uy + xyuz + xz u2 + yz

]

.

2. 34

x2 + yz = 9

(u+ x) y = 4

(u+ x) z = 8

u2 + yz = 9

pi u+ x 6= 0. y

z=

4

8 z = 2y.

pi

x2 + 2y2 = 9

(u+ x) y = 4

u2 + 2y2 = 9

pi pi 1 3 pi x2 = u2 u 6= x u = x.pi

x2 + 2y2 = 9

2xy = 4

pi

x = 1, y = 2,

x = 1, y = 2,x = 2

2, y =

1

2

2

x = 2

2, y = 12

2

pi pi pipi (x, y, z, u)

u = 1, x = 1, y = 2, z = 4,u = 1, x = 1, y = 2, z = 4,

u = 2

2, x = 2

2, y =1

2

2, z =

2,

u = 2

2, x = 2

2, y = 12

2, z =

2

, A .

2.2.34. pi : (IA)1= I + A + A2 + A3 + ... .pi. (

I + A + A2+A3+....)

(IA) = I + A + A2+A3+....AA2A3+.... = I

.

2. 35

pi +An pi An .(: pi pi pi

Bn= I + A + A2+...+ An

Bn (IA) = IAn+1. pi pipi An 0. ;)

2.3

2.3.1.

A =

[1 12 1

], B =

1 1 32 0 00 2 1

.pi A2 , A3, B2 , B3.

pi.

A2 =

[3 00 3

],A3 =

[3 36 3

],B2 =

1 7 62 2 64 2 1

,B3 = 15 11 32 14 128 2 11

2.3.2. [

3 51 2

]1=

[2 51 3

].

2.3.3. 0 1 30 0 11 2 6

1 = 2 0 11 3 0

0 1 0

.2.3.4. pi pi pi, pi.

A =

[3 51 2

], B =

[4 02 1

], C =

[1 22 4

].

pi.A1 =

[2 51 3

], B1 =

[14

012

1

], C1 pi.

2.3.5. pi pi pi, pi.

A =

4 0 00 2 00 0 3

, B = 0 1 30 0 1

1 2 6

, C = 4 0 00 2 1

0 1 1

.pi.

A1 =

14 0 00 12

00 0 1

3

, B = 2 0 11 3 0

0 1 0

, C = 14 0 00 1

313

0 13

23

.

.

2. 36

2.3.6. 0 1 00 0 11 0 0

1 = 0 0 11 0 0

0 1 0

.pi ; (pi : pi - pi).

2.3.7. 0 1 01 0 00 0 1

. pi ;

2.3.8.

A =

1 0 00 1 00 0 1

.pi.

A1 =

1 0 00 1 00 0 1

.2.3.9. pi : (A1)1= A, (A B)1 = B1 A1, (AT )1 = (A1)T .2.3.10. pi : (IA)1= I + A + A2 + A3 + ... .2.3.11. pi [

i 00 i

]K,

[0 ii 0

]K K.

2.3.12. pi a 0 0 00 b 0 00 0 c 00 0 0 d

K

=

aK 0 0 00 bK 0 00 0 cK 00 0 0 dK

.2.3.13. pi [

a 10 a

]K K.

.

2. 37

2.3.14. pi [A 00 B

]K=

[AK 00 BK

],

pi A B pi.

2.3.15. A N N A2 = I. B =12

(I + A), C =12

(IA). B2= C2 = I BC = 0.

2.3.16. A, B N N pi A1.

(A + B) A1 (AB) = (AB) A1 (A + B) .

2.3.17.

A =

1 1 35 2 62 1 3

3.

2.3.18. : A 2, A (IA)k = A k {1, 2, ...}.2.3.19. pi A,B,C A + B + C = I, - AB = BC = CA = 0.

2.3.20.

AN =

1N

1N

... 1N

1N

1N

... ...... ... ... ...1N

... ... 1N

. k {1, 2, ...} (AN)k = AN .2.3.21.

A =

0 1 14 3 43 3 4

.

2.3.22. A , (I + A) (IA) = 0.2.3.23. pi

A =

[9 48 9

].

pi B = A1/2.

.

3 pi

3.1

3.1.1. A AT pipi A :(

AT)mn

= Anm.

.

A =

r1r2...

rM

AT = [ rT1 rT2 ... rTM ] .3.1.2. pi pi :

(A + B)T= AT+BT , (C D)T= DT CT .3.1.3. pi A AT = A (. A ).

3.1.4. pi A AT = A.3.1.5. pi pi .

1. A + AT AAT .2. A =1

2 (A + AT )+ 1

2 (AAT ).

3.1.6. pi A N N , (m = 1, 2, ..., N , n = 1, 2, ..., N ) m 6= n amn = 0. .

A =

a11 0 0 ... 00 a22 0 ... ...0 0 a33 ... ...... ... ... ... ...0 ... ... ... aNN

.

38

.

3. 39

3.1.7. pi A N N , ( m = 1, 2, ..., N , n = 1, 2, ..., N ) m > n amn = 0. .

A =

a11 a12 a13 ... a1N0 a22 a23 ... ...0 0 a33 ... ...... ... ... ... ...0 ... ... ... aNN

.3.1.8. pi A N N , ( m = 1, 2, ..., N , n = 1, 2, ..., N ) m < n amn = 0. .

A =

a11 0 0 ... 0a21 a22 0 ... ...a31 a32 a33 ... ...... ... ... ... ...aN1 ... ... ... aNN

.3.1.9. MN pi A pi pi,

A =

A11 A12 ... A1LA21 A22 ... ...... ... ... ...

AK1 ... ... AKL

pi Akl (k = 1, 2, ..., K l = 1, 2, ..., L) pi Mk Nl M1 +...+MK = M , N1 + ...+NL = N .

3.1.10. pi A,B:

A =

A11 A12 ... A1LA21 A22 ... ...... ... ... ...

AK1 ... ... AKL

, B =

B11 B12 ... B1LB21 B22 ... ...... ... ... ...

BK1 ... ... BKL

pi Akl Bkl k = 1, 2, ..., K l = 1, 2, ..., L.

A + B =

A11 + B11 A12 + B12 ... A1L + BLA21 + B21 A22 + B22 ... ...

... ... ... ...AK1 + BK1 ... ... AKL + BKL

.3.1.11. pi A,B:

A =

A11 A12 ... A1LA21 A22 ... ...... ... ... ...

AK1 ... ... AKL

, B =

B11 B12 ... B1LB21 B22 ... ...... ... ... ...

BK1 ... ... BKL

.

3. 40

pi Akp Bpl pi pipi- .

A B =

P

p=1 A1p Bp1P

p=1 A1p Bp2 ...P

p=1 A1p BpLPp=1 A2p Bp1

Pp=1 A2p Bp2 ... ...

... ... ... ...Pp=1 AKp Bp1 ... ...

Pp=1 AKp BpL

. pi pi pipi A B pi-pi A,B pi pipi pi.

3.1.12. N N pi A , .

A =

A11 0 ... 00 A22 ... ...... ... ... ...0 ... ... AKK

Akk pi ( k {1, 2, ..., K})

A1=

A111 0 ... 00 A122 ... ...... ... ... ...0 ... ... A1KK

3.2

3.2.1. pi pi

A =

1 24 16 7

, B = [ a bc d

]pi.

AT =

[1 4 62 1 7

], BT =

[a cb d

].

3.2.2. pi pi pi pi ;

A =

[3 22 1

], B =

[3 00 5

], C =

3 3 43 1 24 2 1

, D = 0 3 43 0 24 2 0

,E =

[3 6 16 3 2

], F =

3 66 31 5

, G = [ 3 66 3

], H =

[0 66 0

].

pi. A,B,G D,H .

.

3. 41

3.2.3. x, y, z pi

A =

2 x z22x y 4 yy 1 1

.

pi. pipi

2x y = x, y = z2, y = 1.

pi x = 1, y = 1, z = 1. . pi 2 1 11 4 11 1 1

.3.2.4. pi : A B pi, AB AB = BA.

pi. AB pipi (AB)T = AB. (AB)T = BTAT = BA. . pipi AB = BA. ,AB = BA AB = BTAT = (AB)T , . AB .3.2.5. pi (A + B)T= AT+BT .

pi. pi A,B M N . ,((A + B)T

)mn

= (A + B)nm = anm + bnm =(AT)mn

+(BT)mn

m {1, 2, ...,M} n {1, 2, ..., N}. pi (A + B)T =AT + BT .

3.2.6. pi AAT (pi A ).

pi. ..

A =

[2 13 4

].

AT =

[2 31 4

] AAT =

[2 13 4

][

2 31 4

]=

[0 44 0

]pi .

3.2.7. pi A AT (pi A -).

pi.(AAT )T = AT (AT )T = ATA = (AAT ). , -

B = AAT , BT = B, pi B = AAT .

.

3. 42

3.2.8. pi pi A pi pi.

pi. pi pi C piD

A = C + D. (3.1)

AT= CT+DT = CD. (3.2) (3.1) (3.2)

C =1

2

(A + AT

), D =

1

2

(AAT )

.A =

1

2

(A + AT

)+

1

2

(AAT )

pi 3.1.5.3.2.9. pi (AB)T = BTAT .

pi. ..

A =

[2 1 33 4 5

], B =

2 11 34 5

.

AB =

[2 1 33 4 5

] 2 11 34 5

= [ 15 1430 40

]

BTAT =

[2 1 41 3 5

] 2 31 43 5

= [ 15 3014 40

]= (AB)T .

3.2.10. pi (AB)T = BTAT .pi. A M K B K N . (

(AB)T)mn

= (AB)nm =Kk=1

ankbkm =Kk=1

bkmank =Kk=1

(BT)mk

(AT)kn

=(BTAT

)mn.

m {1, 2, ...,M} n {1, 2, ..., N}, (AB)T =BTAT .3.2.11. pi pi pi , , - ;

A =

[3 20 1

], B =

[3 00 5

], C =

3 3 00 1 20 0 1

, D = 1 0 03 4 04 2 0

,E =

[3 6 16 3 2

], F =

3 66 31 5

, G = 3 0 20 3 1

0 0 0

, H = 0 0 60 1 06 0 0

.

.

3. 43

pi. A,B,C,G . D . B .

3.2.12. pi

B =

[1 xy 4

].

x, y B2 pi.pi.

B2=

[1 xy 4

]2=

[1 + xy 5x

5y xy + 16

].

x = 0, y , B2 x , y = 0, B2 .

3.2.13. pi

B =

[x+ y x+ 1y2 + 1 4 + y

].

x, y B2 pi.pi.

B2=

[x+ y x+ 1y2 + 1 4 + y

]2=

[(x+ y)2 + (x+ 1) (y2 + 1) (4 + x+ 2y) (x+ 1)

(4 + x+ 2y) (y2 + 1) (x+ 1) (y2 + 1) + (4 + y)2

].

B2 pipi

(4 + x+ 2y)(y2 + 1

)= 0

pi pipi 4 + x + 2y = 0 x = 4 2y. . pi

B (y) =

[ 4 y 3 2yy2 + 1 4 + y

] (B (y))2 ( ;).

B2 pi pi

x+ 1 = 0

pi pipi x = 1. . pi

B (y) =

[ 1 y 0y2 + 1 4 + y

] pi (B (y))2 .

.

3. 44

3.2.14. pi A

A2 =

[1 20 4

].

pi. pi

A =

[x yz u

].

pipi

A2=

[x yz u

]2=

[x2 + yz xy + yuzx+ uz yz + u2

]=

[1 20 4

].

x2 + yz = 1

xy + yu = 2

zx+ uz = 0

yz + u2 = 4

pi pi z = 0 x = u. pi pipi

x2 = 1

y (x+ u) = 2u2 = 4

pi :

x = 1, z = 0, u = 2, y =2

3,

z = 0, u = 2, x = 1, y = 2,x = 1, z = 0, u = 2, y = 2,z = 0, u = 2, x = 1, y = 2

3

pipi

x2 + yz = 1

y 0 = 20 = 0

yz + u2 = 4

pi . pi pi A pi pi :

A1 =

[1 2/30 2

], A1 =

[ 1 20 2

], A1 =

[1 20 2

], A1 =

[ 1 2/30 2

].

.

.

3. 45

3.2.15. 2 2 pi A A2 .pi.

A =

[x yz u

].

A2 =

[x yz u

]2=

[x2 + yz xy + yuzx+ uz yz + u2

] A2 , pipi

y (x+ u) = 0 = 0z (x+ u) = 0.

.

1. x = u, y, z .2. x = u, y = 0, z .3. x = u, z = 0, y .4. y = 0, z = 0, x, u .

pipi 2 3 pipipi 1. pi pi pi pi :

A (x, y, z) =

[x yz x

], A (x, u) =

[x 00 u

].

3.2.16. pi pi.

pi. pi A,B N N . . m {1, 2, ..., N}, n {1, 2, ..., N} m < n amn = 0. (m,n) AB m < n.

(AB)mn =Nk=1

amkbkn =mk=1

amkbkn +N

k=m+1

amkbkn.

N

k=m+1 amkbkn amk m < k ( ;) pi amk = 0. m

k=1 amkbkn bmk k m < n ( ;) pi bkn = 0. pi, , (m,n) m < n (AB)mn = 0, . AB .

3.2.17. pi

A =

1 2 34 5 67 8 9

.

3. 46

pi pi

A =

[A11 A12A21 A22

]pi

A11=

[1 24 5

], A12=

[36

], A21=

[7 8

], A22=

[9].

pi, pi

A2 =

[A11A11 + A12A21 A11A12 + A12A22A21A11 + A22A21 A21A12 + A22A22

].

pi.

A2=

1 2 34 5 67 8 9

2 = 30 36 4266 81 96

102 126 150

A11A11+A11A12=

[1 24 5

] [1 24 5

]+

[36

] [7 8

]=

[9 1224 33

]+

[21 2442 48

]=

[30 3666 81

]

A11A12 + A12A22=

[1 24 5

] [36

]+

[36

][9] =

[1542

]+

[2754

]=

[4296

]A21A11+A22A21 =

[7 8

] [ 1 24 5

]+[

7 8] [

9]

=[

39 54]+[

63 72]

=[

102 126]

A21A12 + A22A22 =[

7 8] [ 3

6

]+ [9] [9] = [69] + [81] = [150]

pi [A11A11 + A12A21 A11A12 + A12A22A21A11 + A22A21 A21A12 + A22A22

]=

30 3666 81 4296102 126 150

= A2.3.2.18.

A =

1 2 0 01 4 0 0

0 0 1 30 0 1 2

pi A1.

pi. A ,

A11 =

[1 21 4

], A22 =

[1 31 2

]

.

3. 47

pi 3.1.12 [A11 00 A22

]1=

[A111 00 A122

].

pi 3.1.12

A111 =[

1 21 4

]1=

[231

316

16

], A122 =

[1 31 2

]1=

[ 2 31 1

].

pi

A1 =

231

316

16

0 00 0

0 00 0

2 31 1

.3.2.19. pi , N N pi A , .

A =

A11 0 ... 00 A22 ... ...... ... ... ...0 ... ... AKK

Akk pi ( k {1, 2, ..., K})

A1=

A111 0 ... 00 A122 ... ...... ... ... ...0 ... ... A1KK

pi.

B =

A111 0 ... 00 A122 ... ...... ... ... ...0 ... ... A1KK

pi 3.1.12

A11 0 ... 00 A22 ... ...... ... ... ...0 ... ... AKK

A111 0 ... 00 A122 ... ...... ... ... ...0 ... ... A1KK

=P

p=1 A1p Bp1P

p=1 A1p Bp2 ...P

p=1 A1p BpLPp=1 A2p Bp1

Pp=1 A2p Bp2 ... ...

... ... ... ...Pp=1 AKp Bp1 ... ...

Pp=1 AKp BpL

.

.

3. 48

(1, 1) pi

Pp=1

A1p Bp1 = A11 A111 + 0 0 + ...+ 0 0 = I.

(1, 2) pi

Pp=1

A1p Bp2 = A11 0 + 0 A122 + ...+ 0 0 = 0.

pi pipi AB. A11 0 ... 00 A22 ... ...... ... ... ...0 ... ... AKK

A111 0 ... 00 A122 ... ...... ... ... ...0 ... ... A1KK

=

I 0 ... 00 I ... ...... ... ... ...0 ... ... I

= I. pi

A111 0 ... 00 A122 ... ...... ... ... ...0 ... ... A1KK

A11 0 ... 00 A22 ... ...... ... ... ...0 ... ... AKK

=

I 0 ... 00 I ... ...... ... ... ...0 ... ... I

= Ipi

A1 =

A111 0 ... 00 A122 ... ...... ... ... ...0 ... ... A1KK

6 .

3.3

3.3.1. pi pi.

3.3.2. pi pi pi:

A =

[1 32 4

], B =

[1 6 32 8 4

], C =

123

.pi.

AT=

[1 23 4

], BT=

1 26 83 4

, CT= [ 1 2 3 ] .3.3.3. pi : k {1, 2, 3...} 2 2 pi Ak () - A () .

.

3. 49

3.3.4. pi : N {1, 2, 3...} N N pi A2 () A () .

3.3.5. pi pi

A =

1 2 86 1 16 7 5

, B = a bc de f

pi.

AT =

1 6 62 1 78 1 5

, BT = [ a c eb d f

]3.3.6. pi pi pi pi ;

A =

3 2 22 0 12 1 7

, B = [ 3 40 5

], C =

0 8 98 0 29 2 0

, D = 8 3 43 1 24 2 0

,E =

[1 2 3 74 5 6 8

], F =

[3 66 3

], G =

[0 66 3

], H =

[0 66 0

].

pi. A,F . H .

3.3.7. x, y, z pi

A =

2 x2 z2x y 4 y0 1 1

.

pi. x = 1, y = 1, z = 0.

3.3.8. pi pi pi , , - ;

A =

3 0 04 5 02 1 6

, B = [ 4 70 5

], C =

3 3 40 1 20 2 1

, D = 0 3 00 0 2

0 0 0

,E =

[3 6 16 3 2

], F = [3] , G =

[0 66 0

], H =

[0 4 66 4 0

].

pi. A . B,D .

3.3.9. pi , A, AAT .

3.3.10. (A B C ... U)T ;pi. (A B C ... U)T = UT ... CT BT AT .

.

3. 50

3.3.11. pi A,B. pi AB AB = BA.

3.3.12. NN pi A,B , pi A B A B = B A.3.3.13. N N pi A M N pi B, pi BT A B A .

3.3.14. pi

A =

[x y0 3

] x, y A2 pi.

pi. x = 3 y R, y = 3 x R.3.3.15. x, y B2 pi

B =

[5 xy 6

]2 pi.

pi. x = 0, y .

3.3.16. x, y B2 pi pi pi.

pi. x = 0, y = 0.

3.3.17. pi A

A2 =

[1 00 1

].

pi. [1 00 1

],

[ 1 00 1

].

3.3.18. pi A

A2 =

[1 20 1

].

pi. [1 10 1

],

[ 1 10 1

].

3.3.19. pi

C =

[x yz u

].

x, y, z, u C2 .pi. u = x (y = 0 z = 0).

.

3. 51

3.3.20. pi pi.

3.3.21. pi

A =

1 1 42 1 07 0 9

pi pi

A =

[A11 A12A21 A22

]pi

A11=

[1 12 1

], A12=

[40

], A21=

[7 0

], A22=

[9].

pi, pi

A2=


].

3.3.22. A =[

A11 A12A21 A22

], pi A N N , A11 N1 N1

A22 N2 N2 (N = N1 + N2). pi B =[

B11 B12B21 B22

], pi B

N N , B11 N1 N1 B22 N2 N2. pi

AB =

[A11 A12A21 A22

]=

[A11B11+A12B21 A11B12+A12B22A21B11+A22B21 A21B12+A22B22

].

3.3.23. 3 3 pi

A =

a11 a12 a13a21 a22 a23a31 a32 a33

pi pi

A =

[A11 A12A21 A22

]pi

A11=[a11

], A12=

[a12 a13

], A21=

[a21a31

], A22=

[a22 a23a32 a33

].

pi, pi

A2 = =


].

.

3. 52

3.3.24. 1 0 ... 00 2 ... 0... ... ... ...... ... ... N

a11 a12 ... a1Na21 a22 ... a2N... ... ... ...aN1 ... ... aNN

=

1a11 1a12 ... 1a1N2a21 2a22 ... 2a2N... ... ... ...

NaN1 ... ... NaNN

.3.3.25. A,B pi. AB, BA .

3.3.26. pi A

A2 =

[1 10 1

]pi.. pi pi :

A =

[1 1/20 1

] A =

[ 1 1/20 1

].

3.3.27. A,B N N pi, A AB = BA, B .

3.3.28. , K = 1, 2, ... a 1 00 a 10 0 a

K = aK KaK1 (K1)K2 aK20 aK KaK1

0 0 aK

.3.3.29.

A =

1 0 01 0 10 1 0

. K = 1, 2, ... AK+2 = AK + A2 I.3.3.30. , K = 1, 2, ... [ 1 2

4 1

]2K=

[32K 0

0 32K

].

3.3.31. pi

A =

1 0 0 01 1 0 00 1 1 01 1 0 1

. A = I + B, pi B .

.

4

pi pi pi . pi pi pi ! pi pi pi pi : pi pipi, pi pi pi . pi pi pi pi .( pi 14.)

pi pi , pi , .

4.1

4.1.1. N N () pi A , A. D (A) |A| .

1. A 1 1, . A = [a11], |A| = a11.2. A N N N > 1,

|A| = (1)1+1 a11A11 + (1)1+2 a12A12 + ...+ (1)1+N a1NA1N , (4.1)

pi A1n ( n = 1, 2, ..., N ) pipi pi pipi pi A pi n- .

4.1.2. pi N , pi - N 1.4.1.3. (4.1) 2 2 pi A pi a11 a12a21 a22

= a11a22 a12a21. (4.2)

53

.

4. 54

4.1.4. (4.2) pi pipi [ a11 a12a21 a22

] = a11a22 a12a21. (4.3) pipi, pi, pi pi.

4.1.5. (4.1) 3 3 pi A pia11 a12 a13a21 a22 a23a31 a32 a33

= a11a22a33 a11a23a32 a12a21a33 + a13a21a32 + a12a23a31 a13a22a31.4.1.6. pi pi : N N pi , pi pi N , pi pi. pi , pi pi pi pi(4.1).

4.1.7. N N pi A |A| = AT . (4.4)

( pi .)

4.1.8. N N pi A, pi :A =

[a1 a2 ... aN

].

pi .

1. pi, pi : a1 ... am ... an ... aN = a1 ... an ... am ... aN . (4.5)2. an pi an = an+an,

: a1 ... an+an ... aN = a1 ... an ... aN + a1 ... an ... aN .(4.6)

3. pi 1:

|I| = 1. (4.7)

4.1.9. N N pi A, pi :A =

[a1 a2 ... aN

].

pi .

.

4. 55

1. () pi , : a1 ... am ... am ... aN = 0. (4.8)2. pipi () pi , pi-

pi pi : a1 ... an ... aN = a1 ... an ... aN . (4.9)3. () pi pipi ,

pi : a1 ... am+an ... an ... aN = a1 ... am ... an ... aN . (4.10)4. () pi , : a1 ... 0 ... aN = 0. (4.11)

4.1.10. N N pi A ( ) , :

a11 a12 ... a1N0 a22 ... a2N... ... ... ...0 0 ... aNN

=a11 0 ... 0a21 a22 ... 0... ... ... ...aN1 aN2 ... aNN

= a11a22...aNN . (4.12)4.1.11. N N pi A,B.

|AB| = |A| |B| . (4.13)4.1.12. N N pi A pi :

A =

A11 0 ... 00 A22 ... ...... ... ... ...0 ... ... ANN

|A| = |A11| |A22| ... |ANN |4.1.13. N N pi A A1 |A| 6= 0.4.1.14. pi pi A adj (A)

adj (A) =

(1)1+1A11 (1)2+1A21 ... (1)N+1AN1(1)1+2A12 (1)2+2A22 ... (1)N+2AN2... ... ... ...

(1)1+N A1N (1)2+N A2N ... (1)N+N ANN

,pi Amn pi pipi pi pipi pi A - pi m- n- .

.

4. 56

4.1.15. N N pi A |A| 6= 0. A pi pi

A1 =1

|A|adj (A) =1

|A|

(1)1+1A11 (1)2+1A21 ... (1)N+1AN1(1)1+2A12 (1)2+2A22 ... (1)N+2AN2... ... ... ...

(1)1+N A1N (1)2+N A2N ... (1)N+N ANN

. (4.14)

4.2

4.2.1. pi

A =

[1 23 4

].

. pi (4.2)

|A| = a11a22 a12a21 = 1 4 3 3 = 2.4.2.2. pi

A =

1 2 34 5 63 2 0

, B =

1 2 30 5 60 0 4

.pi. A:

|A| =

1 2 34 5 63 2 0

= +1 5 62 0

2 4 63 0+ 3 4 53 2

= 1 (12) 2 (18) + 3 (7) = 3.

B:

|B| =

1 2 30 5 60 0 4

= +1 5 60 4

2 0 60 4+ 3 0 50 0

= 1 20 2 0 + 3 0 = 20.

pi pi |B| = 1 5 4 = 20, B .

4.2.3. pi

A =

0 1 3 20 5 2 03 3 0 00 0 1 6

.

.

4. 57

pi. A AT , pi pi pi pi pi :

|A| =

0 0 3 01 5 3 03 2 0 12 0 0 6

= 0 0 + 3

1 5 03 2 12 0 6

0= 3

2 10 6 3 5 3 12 6

+ 0 = 336.4.2.4. pi

A =

3 2 0 11 5 3 00 0 3 02 0 0 6

.pi. A pi pi pipi pi

pi A:

|A| =

0 0 3 01 5 3 03 2 0 12 0 0 6

= 336.4.2.5. pi

A =

6 0 3 40 5 6 00 0 7 10 0 0 1

.pi. pi , |A| = 6 5 7 1 = 210.

4.2.6. pi

A =

[a11 a12a21 a22

]pi. pi .

|A| = (1)1+1 a11A11 + (1)1+2 a12A12.

A11 a22, pi pi pipi pi A pi pi . A12 a21, pipi pipi pi A pi .

|A| = +a11a22 a12a21.

.

4. 58

4.2.7. pi

A =

a11 a12 a13a21 a22 a23a31 a32 a33

pi.

|A| = (1)1+1 a11A11 + (1)1+2 a12A12 + (1)1+3 a13A13. pi A11, A12, A13 pipi pi 2 2 :

A11 =

a22 a23a32 a33 = a22a33 a23a32,

A12 =

a21 a23a31 a33 = a21a33 a23a31,

A13 =

a21 a22a31 a32 = a21a32 a22a31.

pi

|A| = (1)1+1 a11 (a22a33 a23a32) + (1)1+2 a12 (a21a33 a23a31) + (1)1+3 a13 (a21a32 a22a31)= a11a22a33 a11a23a32 a12a21a33 + a13a21a32 + a12a23a31 a13a22a31.

4.2.8. pi pi pi pi . pi 4.1.7 pi pi pipi pi 22. 4.1.8, 4.1.10,4.1.11 4.1.12, ;;. ( pi 14) pi pi (4.1.1), pi pipi pi 4.11.4.2.9. |A| = AT pi

A =

1 0 23 4 62 5 1

pi.

1 0 23 4 62 5 1

= 1 4 65 1

0 3 62 1+ 2 3 42 5

= 1 (26) + 2 7 = 12

1 3 20 4 52 6 1

= 1 4 56 1

3 0 52 1+2 0 42 6

= 1 (26)3 (10)+2 (8) = 12.1pi 14, pi

pipi , pi pi .

.

4. 59

4.2.10. |A| = AT piA =

[a11 a12a21 a22

],

pi. a11 a12a21 a22 = a11a22 a12a21 a11 a21a12 a22 = a11a22 a21a12

pi .

4.2.11. piA =

[1 23 4

] |A| = |A| pi A pi pi pipi A.

pi. 1 23 4 = 2 2 14 3

= 2.4.2.12. pi

A =

[a11 a12a21 a22

] |A| = |A| pi A pi pi pipi A.

pi. a11 a12a21 a22 = a11a22 a12a21 a12 a11a22 a21

= a12a21 a11a22 = (a11a22 a12a21) .4.2.13. pi

A=[

1 23 4

], A=

[1 53 6

], A =

[1 2+ 53 4+ 6

],

|A| = |A|+ |A|.pi. 1 23 4

= 2, 1 53 6 = 9

1 2+ 53 4+ 6 = 1 (4+ 6) 3 (2+ 5)

= (1 4 3 2) + (1 6 3 5) = 1 23 4

+ 1 53 6 .

.

4. 60

4.2.14. pi a11 a12 + a12a21 a22 + a22 = a11 a12a21 a22

+ a11 a12a21 a22 .

pi. , pi pi

a11 a12a21 a22

= (a11a22 a12a21) , a11 a12a21 a22

= (a11a22 a12a21) , a11 a12 + a12a21 a22 + a22 = a11a22 + a11a22 a21a12 a21a12

= (a11a22 a21a12) + (a11a22 a21a12) .

4.2.15. 1 0 00 1 00 0 1

= 1.pi.

1 0 00 1 00 0 1

= 1 1 00 1

0 0 00 1+ 0 0 00 1

= 1 (1 1 0 0) 0 + 0 = 1.

pi.

4.2.16. pi pi 0.

pi. .. 1 2 12 0 23 5 3

= 1 0 25 3

2 2 23 3+ 1 2 03 5

= 1 (10) 2 (6 6) + 1 10 = 0.

4.2.17. pi pi 0.pi. pi pi 4.1.8.

pi ,

|a1...am...am...aN | = |a1...am...am...aN | |a1...am...am...aN | = 0.

. pi ( ) pi pipi pi, .

.

4. 61

4.2.18. pi pipi pi , pipi pi .

pi. .. 1 2 12 0 23 5 4

= 1 0 25 4

2 2 23 4+ 1 2 03 5

= 4

1 2 12 0 23 5 4

= 1 0 25 4

2 2 23 4+ 1 2 03 5

= 10 4+ 10 = 4.


pi. pi 4.1.8. pi pi n- an =an + a

n, pi = 0,

|a1... an + 0 an...aN | = |a1...an...aN |+ 0 |a1...an...aN | = |a1...an...aN | .. pi ( ) pi pipi pi, .

4.2.20. pi pi , pi .

pi. .. 1 2 12 0 23 5 4

= 1 0 25 4

2 2 23 4+ 1 2 03 5

= 4

1 2 + 1 12 0 + 2 23 5 + 3 4

= 1 0 + 2 25 + 3 4

(2 + ) 2 23 4+ 1 2 0 + 2 3 5 +

= (10 6) (2 + ) 2 + (10 + 2) = 4.


pi. pi 4.1.8. pi

[a1...am...an...aN ]

pi

[a1...1 am + an...an...aN ]

.

4. 62

|a1...1 am + an...an...aN | = 1 |a1...am...an...aN |+ |a1...an...an...aN |= |a1...am...an...aN |

|a1...an...an...aN | = 0 ( pi ).4.2.22. pi pi 0.

pi. .. 1 0 12 0 23 0 4

= 1 0 20 4

0 2 23 4+ 1 2 03 0

= 1 0 0 (8 6) + 1 0 = 0.

4.2.23. pi pi 0.pi. pi 4.1.8.

|a1...0 am + 0 am...aN | = 0 |a1...am...aN |+ 0 |a1...am...aN | = 0.4.2.24.

4 0 0 02 5 0 03 6 8 04 7 9 10

4 5 8 10.

pi. 4 0 0 02 5 0 03 6 8 04 7 9 10

= 4

5 0 06 8 07 9 10

0 + 0 0= 4 5

8 09 10 0 + 0 0

= 4 5 8 |10| 0 + 0 0 = 4 5 8 10.4.2.25. |AB| = |A| |B| pi

A =

1 2 30 1 23 0 2

, B = 1 0 01 1 0

3 0 2

pi.

AB =

1 2 30 1 23 0 2

1 0 01 1 03 0 2

= 12 2 67 1 4

9 0 4

.

4. 63

|A| =

1 2 30 1 23 0 2

= 5, |B| =

1 0 01 1 03 0 2

= 2, |AB| =

12 2 67 1 49 0 4

= 10.4.2.26. |AB| = |A| |B| pi

A =

[a bc d

], B =

[x yz u

]pi.

AB =

[a bc d

] [x yz u

]=

[ax+ bz ay + bucx+ dz cy + du

]

|A| = a bc d

= ad bc, |B| = x yz u = xu yz

|AB| = ax+ bz ay + bucx+ dz cy + du

= axdu+ bzcy aydz bucx, ,

|A| |B| = (ad bc) (xu yz) = axdu+ bzcy aydz bucx.

4.2.27. |A| = |A11| |A22| ... |ANN | pi

A =

2 4 0 01 3 0 00 0 3 60 0 5 2

pi.

2 4 0 01 3 0 00 0 3 60 0 5 2

= 48.pi pi

A11 =

[2 41 3

] |A11| = 2,

A22 =

[3 65 2

] |A22| = 24

pi, pi, .

4.2.28. N N pi A. |A| = N |A| .pi. pipi pi

A = [a1 a2 ... aN ]

.

4. 64

piA = |a1 a2 ... aN |

= |a1 a2 ... aN |= 2 |a1 a2 ... aN |= ...

= N |a1 a2 ... aN | .4.2.29. N N pi A, pi N = 2M + 1. |A| = 0.

pi. A |A| = AT . A , AT= A.

A = [(1) a1 (1) a2 ... (1) aN ]pi, pi pi |A| = (1)2M+1 |A| = |A|. pi

|A| = AT = |A| = (1)2M+1 |A| = |A| pi |A| = 0.4.2.30. N N pi A, AAT = I. |A| = 1 |A| = 1

pi. AAT = |A| AT = |A| |A| = |A|2AAT = |I| = 1 |A|2 = 1 pi .4.2.31. pi pi ( 3 3 pi)

A1 =1

|A|

A11 A21 A31A12 A22 A32A13 A23 A33

pi

A =

4 2 31 0 10 1 1

, B = 4 0 00 2 0

0 0 3

, C = 4 0 00 2 1

0 1 1

.pi. A

|A| =

4 2 31 0 10 1 1

= 5A11 =

0 11 1 = 1, A12 = 1 10 1

= 1, A13 = 1 00 1 = 1,

A21 =

2 31 1 = 1, A22 = 4 30 1

= 4, A23 = 4 20 1 = 4,

A31 =

2 30 1 = 2, A32 = 4 31 1

= 7, A33 = 4 21 0 = 2,

.

4. 65

pi

A1 =1

5

1 1 21 4 71 4 2

B

|B| =

4 0 00 2 00 0 3

= 24B11 = 6, B12 = 0, B13 = 0,

B21 = 0, B22 = 12, B23 = 0,

B31 = 0, B32 = 0, B33 = 0,

pi

B1 =

1/4 0 00 1/2 00 0 1/3

( B1 , pi B, pi ;)

C =

4 0 00 2 10 1 1

4 0 00 2 10 1 1

= 12 2 11 1

= 3, 0 10 1 = 0, 0 20 1

= 0, 0 01 1 = 0, 4 00 1

= 4, 4 00 1 = 4, 0 02 1

= 0, 4 00 1 = 4, 4 00 2

= 8,pi 4 0 00 2 1

0 1 1

1 = 14 0 00 1

313

0 13

23

. pi pipi[

13

131

323

],

[2 11 1

] .

.

4. 66

4.2.32. pi |A| = 0 pi A1.pi. |A| = 0 pi A1. |A B| =

|A| |B|, pi 1 = |I| = A A1 = 0 A1 = 0

pi pi.

4.2.33. N N pi A pi |3A| = 27 |A| . N ;

pi. pi 4.2.28 |3A| = 3N |A|. 3N |A| = 27 |A|

pi N = 3 |A| ( pipi pi pi- N ).

4.2.34. pi pi x 1 x2x 2x+ 1 .

pi. pi

f (x) =

1 x2x 2x+ 1 = 2x+ 1 x3

df/dx = 2 3x2. ddx1 x2ddxx 2x+ 1

+ 1 ddxx2x ddx

(2x+ 1)

= 0 x21 2x+ 1+ 1 2xx 2

= x2 + 2 2x2 = 2 3x2.pi pipi pi ;

4.2.35. pi pi x sinx x3 x2

x ex 2x+ 1x+ 1 0 1

.pi. pi

sinx x3 x2

x ex 2x+ 1x+ 1 0 1

= (sinx x3 x2) ex + x3 (2x2 + 2x+ 1)

D =d

dx

sinx x3 x2

x ex 2x+ 1x+ 1 0 1

= (cosx 4x2 2x+ (sinx) x3) ex + 10x4 + 8x3 + 3x2pi

.

4. 67

4.2.36.

D1 =

ddx

sinx x3 x2ddxx ex 2x+ 1

ddx

(x+ 1) 0 1

=

cosx x3 x2

1 ex 2x+ 11 0 1

= ex cosx+ 2x4 x2exD2 =

sinx d

dxx3 x2

x ddxex 2x+ 1

x+ 1 ddx

0 1

=

sinx 3x2 x2

x ex 2x+ 1x+ 1 0 1

= (sinx) ex + 6x3 + 6x4 x3ex + 3x2 x2exD3 =

sinx x3 d

dxx2

x ex ddx

(2x+ 1)x+ 1 0 d

dx1

=

sinx x3 2xx ex 2

x+ 1 0 0

= (x+ 1) (2x3 2exx) . pi pi pi

D = D1 +D2 +D3.

pi pipi pi ;

4.2.37. b+ c a b ac+ a b c ba+ b c a c

= 3abc a3 b3 c3.pi. pi pi-

pi:b+ c a b ac+ a b c ba+ b c a c

=b+ c b ac+ a c ba+ b a c

=c b aa c bb a c

= c

(c2 + ab)+ b (ac b2)+ a (a2 + bc)= 3abc a3 b3 c3.

4.2.38. a b c 2a 2a

2b b c a 2b2c 2c c a b

= (a+ b+ c)3pi. pi pi :

a b c 2a 2a2b b c a 2b2c 2c c a b

=a+ b+ c a+ b+ c a+ b+ c

2b b c a 2b2c 2c c a b

= (a+ b+ c)

1 1 12b b c a 2b2c 2c c a b

.

.

4. 68

pi pi:

(a+ b+ c)

1 1 12b b c a 2b2c 2c c a b

= (a+ b+ c)

1 0 02b c a b 02c 0 a b c

= (a+ b+ c) (a+ b+ c) (a+ b+ c) .

4.2.39. pi x a a aa x a aa a x aa a a x

.pi. pi pi, pi pi:

x a a aa x a aa a x aa a a x

=x a x a x a xa x a 0 0a 0 x a 0a 0 0 x a

. , pi (x a):

x a x a x a xa x a 0 0a 0 x x 0a 0 0 x a

= (x a)3

x 1 1 1a 1 0 0a 0 1 0a 0 0 1

=(x a)3

a 0 0a 1 0a 0 1

+x 1 1a 1 0a 0 1

= (x a)3 (3a+ x) .

4.2.40. N N pi A

AN =

1 1 ... 11 0 ... ...... ... ... ...1 ... ... 0

|AN | = (1)N1 .

pi. pi pi pi pi :

|AN | =

1 1 1 ... 11 0 1 ... ...... ... ... ... ...1 ... ... ... 0

=

0 1 0 ... 01 0 1 ... ...... ... ... ... ...1 ... ... ... 0

=

1 1 1 ... 11 0 1 ... ...... ... ... ... ...1 ... ... ... 0

= |AN1| .. |AN | = |AN1| pi |AN1| N 1. pi N 1 pipi .

.

4. 69

4.2.41. N N pi A

A =

1 1 ... 1

1 1 ... ...... ... ... ...1 ... ... 1

|A| = (1)N1 2N1 (N 2) .

pi. pi pi pi pi :

|AN | =

1 1 ... 1

1 1 ... ...... ... ... ...1 ... ... 1

= 21 1 ... 1

1 1 ... ...... ... ... ...1 ... ... 1

2

1 1 ... 11 1 ... ...... ... ... ...1 ... ... 1

= 2 |AN1| 2 |BN1| ,

pi

|BN1| =

1 1 ... 11 1 ... ...... ... ... ...1 ... ... 1

., pi pi |BN1| pi pi :

|BN1| =

1 1 ... 11 1 ... ...... ... ... ...1 ... ... 1

=

0 2 ... 01 1 ... ...... ... ... ...1 ... ... 1

= 2

1 1 ... 11 1 ... ...... ... ... ...1 ... ... 1

= 2 |BN2| . pi, |BN1| = (2)N2 |B1| = (2)N2, pi |AN |

|AN | = 2 |AN1|+ (2)N1 . (4.15) (4.15) |AN | = (1)N1 2N1 (N 2). pi pi , pi N = 2, 3, ... .

4.2.42.

D (x, y, z) =

x2 x 1y2 y 1z2 z 1

= (x y) (x z) (y z) .pi. pi pi pi

pipi pi. pi pi. D (x, y, z) 3 ,

.

4. 70

pi pi , D (x, y, z) pi pi x ( pi pi y z). , x = y, D (x, y, z) . D (x, x, z) = 0, pi D (x, y, z) pi (x y) (x z). D (x, y, z) pi y pi (y z) pipi D (x, y, z). pi, D (x, y, z) pi x, y, z, pipi .

D (x, y, z) = (x y) (x z) (y z) D (x, y, z) = (x y) (x z) (y z) .

pi x2y pi pipi pi ( ; pi !), pi D (x, y, z) = (x y) (x z) (y z).

pi pi

D (x1, x2, ..., xN) =

xN11 x

N21 ... 1

xN12 xN22 ... 1

... ... ... ...xN1N x

N2N ... 1

.4.2.43. pi

DN (x) =

1 1 1 ... 11 1 x 1 ... 11 1 1 2x ... 1... ... ... ... ...1 1 1 ... N 1 x

. (4.16)

pi. N , DN (x) pi N 1 pi x DN = 0 N 1 N 1 . , x = 0, 1, ..., N 2, , , ..., N- pi. N 1 x1 = 0,x2 = 1, ..., xN1 = N 2. pi

DN (x) = x (x 1) ... (xN + 2) .

4.3

4.3.1. pi pi .

1 23 4 ,

3 0 0

99 1 2373 3 4

,

1 2 31 0 2

3 1 0

,

1 2 32 4 63 1 0

,

0 1 40 2 50 3 6

,

3 2 30 4 20 0 5

.pi. 2,6, 7, 0, 24, 60.

.

4. 71

4.3.2. pi a11 0 ... 00 a22 ... 0... ... ... ...0 0 ... aNN

=a11 0 ... 0a21 a22 ... 0... ... ... ...aN1 aN2 ... aNN

= a11a22...aNN .4.3.3. pi a11 a12a21 a22

+ b11 b12b21 b22 6= a11 + b11 a12 + b12a21 + b21 a22 + b22

.4.3.4. pi a11 a12a21 a22

+ a11 b12a21 b22 = a11 a12 + b12a21 a22 + b22

.4.3.5. pi a11 a12a21 a22

= a11 a12a21 a22 .

a11 a12 a21 a22 ;

4.3.6. a11 a12 a13a21 a22 a23a31 a32 a33

= (a21

a12 a13a32 a33 a22 a11 a13a31 a33

+ a23 a11 a12a31 a32)

4.3.7. pi x y + z z

x+ 1 y + z + 2 z + 11 2 1

= 0.4.3.8. pi

1 + a 1 11 1 + a 11 1 1 + a

= 3a2 + a3.4.3.9. pi

a b 0 0c d 0 00 0 e f0 0 g h

= a bc d

e fg h .

4.3.10. pi 0 i 0 0i 0 0 00 0 0 10 0 1 0

= 1.

.

4. 72

4.3.11. pi 1 1 1a b c

b+ c c+ a a+ b

= 0.4.3.12. pi N {2, 3, ...}

0 1 ... 11 0 ... 1... ... ... ...1 1 .. 0

= (1)N1 (N 1) .

4.3.13. pi b+ c a b ac+ a b c ba+ b c a c

= 3abc a3 b3 c3.4.3.14. pi

a b c db a d cc d a bd c b a

= (a+ b+ c+ d) (a b+ c d) (a b c+ d) (a+ b c d) .4.3.15. pi

1 1 1

1 1 11 1 1

= 22,1 1 1 1

1 1 1 11 1 1 11 1 1 1

= 24.

pi pi N N ;4.3.16. pi

0 a b c da 0 a b cb a 0 a bc b a 0 ad c b a 0

= 0.

4.3.17. pi a a a aa b b ba b c ca b c d

= a (b c) (d c) (a b) .

.

4. 73

4.3.18. M M . pi A, C pi A 0B C = |A| |C| .

4.3.19. pi pi x 1 x2x 2x+ 1 .

pi. 2 3x2.4.3.20. pi pi x

x+ 2 x ex

x+ 1 ex x+ 3ex x 7 x4

pi. 7x4ex + x5ex 3x2 + 4x+ 29 + 8x3ex 6x5 + 2exx2 + exx 10ex 5x4 3e3x.

4.3.21. pi A aij = aji (pi z z),pi |A| pi .4.3.22. pi

(I + A)1 = I (I + A)1 A (I + PQ)1 P = P (I + QP)1 .

4.3.23. pi

(A + UCV)1 = A1 A1 (C1 + VAU)1 VA1.4.3.24. N N pi A

A =

0 1 ... 11 0 ... ...... ... ... ...1 ... ... 0

|A| = (1)N1 (N 1).4.3.25. N N pi A

A =

a+ b a ... aa a+ b ... ...... ... ... ...a ... ... a+ b

|A| = bN1 (Na+ b).

.

4. 74

4.3.26. 0 a1 b1 c1 d1a1 0 a2 b2 c2b1 a2 0 a3 b3c1 b2 a3 0 a4d1 c2 b3 a4 0

= 0.

4.3.27. 1 + a1 a2 a3 ... aNa1 1 + a2 a3 ... aNa1 a2 1 + a3 ... aN... ... ... ... ...a1 a2 a3 ... 1 + aN

= 1 + a1 + a2 + ...+ aN .

4.3.28. pi 2N a 0 ... 0 b0 a ... b 0... ... ... ... ...0 b ... a 0b 0 ... 0 a

=(a2 b2) ..

4.3.29. a b a b1b a b ac d c dd c d c

= 4 (a2 + b2

) (c2 + d2) .4.3.30.

1 + x 1 1 11 1 x 1 11 1 1 + z 11 1 1 1 z

= x2z2.

4.3.31. 0 1 1 11 0 x y1 x 0 z1 y z 0

= x2 + y2 + z2 2 (xy + yz + zx) .

4.3.32. N N pi A pi pi :Amn = a m = n, Amn = b m 6= n.

() pi A1 (A1)mn = x m = n (A1)mn = y m 6= n() a, b pi A1.

.

4. 75

4.3.33. 0 1 2 ... N 11 0 1 ... N 22 1 0 ... N 3... ... ... ... ...

N 1 N 2 N 3 ... 0

= (1)N1 2N2 (N 1) .

4.3.34. 0 1 1 1 11 0 1 1 11 1 0 1 11 1 1 0 11 1 1 1 0

= 4 (1)4

N- .

4.3.35. 1 1 1 1 11 0 1 1 11 1 0 1 11 1 1 0 11 1 1 1 0

= (1)4

N- .

4.3.36. a 1 1 1 ... 11 a 1 1 ... a1 1 a 1 ... 1... ... ... ... ... ...1 1 1 1 ... a

= (a 1)N1 (a+N 1) .

4.3.37. a1a2....aN 6= 0. 1 + a1 1 1 1 ... 1

1 1 + a2 1 1 ... a1 1 1 + a3 1 ... 1... ... ... ... ... ...1 1 1 1 ... 1 + aN

= a1a2...aN

(1 +

1

a1+ ...+

1

aN

).

4.3.38. N N pi

AN=

a 1 0 0 ... 01 a 1 0 ... 00 1 a 1 ... 0... ... ... ... ... ...0 0 0 0 ... a

. |AN | = a |AN1| |AN2|. pi |AN |.

.

4. 76

4.3.39. (a2 + a3)

2 a21 a22

a22 (a3 + a1)2 a22

a23 a23 (a1 + a2)

2

= 2a1a2a3 (a1 + a2 + a3)2 . N- .

4.3.40. x1 + y1 y1 + z1 z1 + x1x2 + y2 y2 + z2 z2 + x2x3 + y3 y3 + z3 z3 + x3

= 2 x1 y1 z1x2 y2 z2x3 y3 z3

.4.3.41.

x a b xa x x bb x x ax b a x

= (a+ b 2x) (a+ b+ 2x) (b a)2 .

4.3.42.

1n=1

1n=1

1n=1 ...

1n=11

n=1

2n=1

2n=1 ...

2n=11

n=1

2n=1

3n=1 ...

3n=1

... ... ... ... ...1n=1

2n=1

3n=1 ...

Nn=1

= N !.

4.3.43. pi 1 a 00 1 a0 0 1

1 = 1 a a20 1 a

0 0 1

. N N pi.4.3.44. pi a 1 11 a 1

1 1 a

1 = a+1a2+a2 1a2+a2 1a2+a2 1

a2+a2a+1

a2+a2 1a2+a2 1a2+a2 1a2+a2 a+1a2+a2

. N N pi.4.3.45. N N pi A I + A + A2+...+ AK= 0. pi A1 = AK .

.

5

5.1

5.1.1. N N

a11x1 + a12x2 + ...+ a1NxN = b1

a21x1 + a22x2 + ...+ a2NxN = b2 (5.1)...

aN1x1 + aN2x2 + ...+ aNNxN = bN

pi a11 a12 ... a1Na21 a22 ... a2N... ... ... ...aN1 aN2 ... aNN

x1x2...xN

=b1b2...bN

(5.2)

Ax = b. (5.3)

pi

A =

a11 a12 ... a1Na21 a22 ... a2N... ... ... ...aN1 aN2 ... aNN

, x =x1x2...xN

, b =b1b2...bN

.5.1.2. |A| 6= 0, (5.3)

x = A1b. (5.4)

77

.

5. 78

5.1.3. ( Cramer) |A| 6= 0, (5.3) :

x1 =

b1 a12 ... a1Nb2 a22 ... a2N... ... ... ...bN aN2 ... aNN


, x2 =

a11 b1 ... a1Na21 b2 ... a2N... ... ... ...aN1 bN ... aNN


, ..., xN =

a11 a12 ... b1a21 a22 ... b2... ... ... ...aN1 aN2 ... bN


.

(5.5)

5.1.4. A M N , M < N . Ax = b, .a11 a12 ... a1Na21 a22 ... a2N... ... ... ...aM1 aM2 ... aMN

x1x2...xN

=

b1b2...bM

(5.6) pi pi . ,pi Cramer, .

a11 a12 ... a1Ma21 a22 ... a2M... ... ... ...aM1 aM2 ... aMM

x1x2...xM

=

b1b2...

bM

(5.7)pi

b1b2...

bM

=

b1b2...bM

a1,M+1 a1,M+2 ... a1,Na2,M+1 a2,M+2 ... a2,N... ... ... ...aM,M+1 aM,M+2 ... aM,N

xM+1xM+2...xN

. (5.8)pi pi (5.7),

a11 a12 ... a1Ma21 a22 ... a2M... ... ... ...aM1 aM2 ... aMM

6= 0, pi x1, x2, ..., xM xM+1, ..., xN . pi pi , pi pi xM+1, xM+2,xN , pi pi pi .

5.1.5. Ax = b .

1. |A| 6= 0, pi pi Cramer. pipi pi (b = 0) : x1 = x2 = ... = xN = 0.

.

5. 79

2. |A| = 0 b = 0 ( ) pi , pi .

3. |A| = 0 b 6= 0 ( ) pi .

5.1.6. Ax = 0 |A| = 0.

5.2

5.2.1. pi , pi A1, Ax = b x = A1b.pi. A1 pi

Ax = b A1Ax = A1b Ix = A1b x = A1b.5.2.2. pi

x1 + 2x2 = 12x1 x2 = 3

pi. Ax = b pi x = A1b.

A =

[1 22 1

], b =

[13

], A1 =

[15

25

251

5

],

x =

[x1x2

]=

[15

25

251

5

] [13

]=

[ 11

].

5.2.3. pi

x1 + x2 = 5x1 x2 = 1 .

pi.

A =

[1 11 1

], b =

[51

], A1 =

[12

12

121

2

],

x =

[x1x2

]=

[12

12

121

2

] [51

]=

[23

].

5.2.4. pi

x1 + 2x2 = 12x1 + 4x2 = 0

pi.

|A| = 1 22 4

= 0,pi A1 pi pi pi.

.

5. 80

5.2.5. pi

x1 + 2x2 + x3 = 22x1 x2 + 2x3 = 1

2x1 + x2 + x3 = 1

pi.

A =

1 2 12 1 22 1 1

, b = 21

1

, A1 = 35 15 12

51

50

45

351

,x =

x1x2x3

= 35 15 12

51

50

45

351

211

= 01

0

.5.2.6. pi

x1 + 2x2 + x3 = 02x1 x2 + 2x3 = 0x1 + x2 + x3 = 0

.

A =

1 2 12 1 21 1 1

|A| =

1 2 12 1 21 1 1

= 0 A1 pi pi .

5.2.7. pi Cramer ( 5.1.3).pi. An pi pi pipi pi A

i- b. , pi n-

|An| =Nk=1

bkAkn. (5.9)

( pi An A Akn ;). pi x Cramer Ax = b. pi ( n = 1, 2, ..., N )

xn =1

|A|Nk=1

bkAkn (5.10)

pipi (pi (5.9) ) i = 1, 2, .., N :

Nn=1

ainxn =Nn=1

ain|A|

Nk=1

bkAkn =1

|A|Nk=1

(Nn=1

ainAkn

)bk.

.

5. 81

Nn=1

ainAkn =

{ |A| i = k,0 i 6= k.

N

n=1 ainAin = |A| pi pi pi pi i- .

Nn=1 ainAkn = 0, i 6= k, pi pi (pi ;)

pi . x (5.10) ( !) Ax = b.

5.2.8. pi (pi ) pi Cramer.

x1 + 2x2 = 12x1 x2 = 3 ,

x1 + x2 = 5x1 x2 = 1 ,

x1 + 2x2 = 12x1 + 4x2 = 0

pi. x1 + 2x2 = 1

2x1 x2 = 3

x1 =

1 23 1 1 22 1 = 1, x2 =

1 12 3 1 22 1 = 1.

x1 + x2 = 5

x1 x2 = 1

x1 =

5 11 1 1 11 1 = 2, x2 =

1 51 1 1 11 1 = 3.

x1 + 2x2 = 12x1 + 4x2 = 0

pi 1 22 4 = 0,

pi Cramer . pi pi , .

5.2.9. pi pi Cramer.

x1 + 2x2 + x3 = 22x1 x2 + 2x3 = 1

2x1 + x2 + x3 = 1. (5.11)

.

5. 82

pi.

x1 =

2 2 11 1 2

1 1 1

1 2 12 1 22 1 1

=

0

5= 0, x2 =

1 2 12 1 22 1 1

1 2 12 1 22 1 1

=

5

5= 1, x3 =

1 2 22 1 12 1 1

1 2 12 1 22 1 1

=

0

5= 0.

5.2.10. pi pi Cramer.x1 + 2x2 + x3 = 02x1 x2 + 2x3 = 0x1 + x2 + x3 = 0

.

pi. 1 2 12 1 21 1 1

= 0 pi Cramer. pi . pi pi-

x1 + 2x2 + x3 = 02x1 x2 + 2x3 = 0

pi x1 + 2x2 = x32x1 x2 = 2x3 .

x1 =

x3 22x3 1 1 22 1

= x3, x2 = 1 x32 2x3

1 22 1 = 0.

pi . , x1, x2 (5.11),

x1 + x2 + x3 = x3 + 0 + x3 = 0.pi

x =

x31x3

= t 10

1

. x3 = t pi pi ( x3). pi :

x = t 10

1

.

.

5. 83

5.2.11. pi pi Cramer.

x1 + 2x2 + x3 = 22x1 x2 + 2x3 = 1x1 + x2 + x3 = 1

.

pi. 1 2 12 1 21 1 1

= 0 pi Cramer . pi pi-

x1 + 2x2 + x3 = 22x1 x2 + 2x3 = 1

pi x1 + 2x2 = 2 x3

2x1 x2 = 1 2x3 .

x1 =

2 x3 21 2x3 1 1 22 1

= x3, x2 = 1 2 x32 1 2x3

1 22 1 = 1.

pi .

x1 + x2 + x3 = x3 + 1 + x3 = 1.

pi

x =

010

+ t 10

1

t R (pi ). pi ; ;

5.2.12. pi

A =

1 2 31 0 10 1 1

Cramer.

.

5. 84

pi.

A1 =

x11 x12 x13x21 x22 x23x31 x32 x33

pipi 1 2 31 0 1

0 1 1

x11 x12 x13x21 x22 x23x31 x32 x33

= 1 0 00 1 0

0 0 1

pi pi pi: 1 2 31 0 1

0 1 1

x11x21x31

= 10

0

, 1 2 31 0 1

0 1 1

x12x22x32

= 01

0

, 1 2 31 0 1

0 1 1

x13x23x33

= 00

1

. pi

x11 =

1 2 30 0 10 1 1

1 2 31 0 1

0 1 1

=

1

2, x21 =

1 1 31 0 1

0 0 1

1 2 31 0 1

0 1 1

= 1

2, x31 =

1 2 11 0 0

0 1 0

1 2 31 0 1

0 1 1

=

1

2

pi pi x21, x22, ..., x33 pi 1 2 31 0 10 1 1

1 = 12 12 11

21

22

12

121

.5.2.13. pi 5.1.5.

pi. |A| 6= 0, pi () A1 Ax = b x = A1b. b = 0, x = A10 = 0.

|A| = 0 b = 0, pipi A, A, pi . pi pipi pi pipi A (pi !) . pi A, pi, pi xn1, xn2, ..., pi. pi , pi A0 = b = 0.

.

5. 85

|A| = 0 b 6= 0 Ax = b u, .Au = b.

Ax = 0 pi pi ( |A| = 0). pi , x x , x 6= x.

Au = bAx = 0

} A (u + x) = b + 0 = b,

. z= u + x Ax = b pi z= u + x Ax = b, z ( x 6= x x + b 6= x + b)., pi pi Ax = 0 pi Ax = b . , Ax = b , pi Ax = b .5.2.14. 5.1.5 2 2.

pi. 2 2 a11x1 + a12x2 = b1 (5.12)a21x1 + a22x2 = b2 (5.13)

(5.12)-(5.13) . pi pi . , a11 a12a21 a22

6= 0 a11a21 6= a12a22pi pi, . . pi a11 a12a21 a22

= 0 a11a21 = a12a22pi pi ( ), pipi ( pi ). pipi pi b1 = b2 = 0, pi pi (0, 0) pipi , pi pi (0, 0) pipi . pi .5.2.15. 5.1.5 3 3.

pi. 3 3 a11x1 + a12x2 + a13x3 = b1 (5.14)a21x1 + a22x2 + a23x3 = b2 (5.15)a31x1 + a32x2 + a33x3 = b3 (5.16)

pipi. pi pipi pi 2 2 , pi pipi : () pipi pi pipi (|A| 6= 0, ), () pipi pipi pi pi (|A| = 0,pi ), () pipi pi (|A| = 0, ).

.

5. 86

5.2.16. pi Ax = 0 |A| = 0.

pi. Ax = 0 pi . , pi . pi 5.1.5 pi : x Ax= 0, R

A (x) = Ax= 0 = 0,. pi ( R). pi 5.1.5 pi |A| = 0.5.2.17. pi

x1 + x2 + x3 = 1

x1 + x2 + x3 = 2

x1 + x2 + x3 = 3

.pi. Ax = b,

|A| = 1 11 11 1

= 3 3+ 2 = (+ 2) ( 1)2 . , pipi |A| = (+ 2) ( 1)2 6= 0, R {2, 1}.5.2.18. pi

x1 + 2x2 = 0

3x1 + x2 = 2

pi .pi. Ax = b,

|A| = 23

= 2 6. , pi |A| = 26 = 0, . = 6.

5.3


x1 + 2x2 = 12x1 x2 = 3 ,

x1 + x2 = 5x1 x2 = 1 ,

2x1 + x2 = 5x1 2x2 = 1

pi. (x1 = 1, x2 = 1), (x1 = 2, x2 = 3), (x1 = 95 , x2 = 75).

.

5. 87


x1 + 3x2 x3 = 23x2 + 2x3 x4 = 5

x3 + x4 = 1x4 = 2

,x1 + 2x2 + x3 = 2

2x1 x2 + 2x3 = 1x1 + x2 + x3 = 1

,x1 + 3x2 x3 = 2

3x1 x2 = 5x1 + x2 x3 = 1

.

pi. (x4 = 2, x3 = 1, x2 = 3, x1 = 8), (x2 = 1, x3 = x3, x1 = x3), x1 = 116 , x2 =12, x3 =

43).


x1 + x2 2x3 = 5x1 x2 + x3 = 12x1 + x2 + x3 = 4

,x1 + x2 2x3 = 5

x1 + x3 = 12x1 + x2 + x3 = 4

,x1 + x2 x3 = 1x1 x2 + x3 = 1x1 + x2 + x3 = 1

.

pi.(x1 =

113, x2 = 8, x3 =

103

), (x1 = 13 , x2 = 4, x3 = 23 ), (x1 = 1, x2 = 1, x3 = 1).


x1 + 3x2 4x3 = 0x1 + x2 x3 = 0

2x1 + 4x2 5x3 = 0,

x1 + 3x2 4x3 = 1x1 + x2 x3 = 3

2x1 + 4x2 5x3 = 4,

x1 + 3x2 4x3 = 1x1 + x2 x3 = 2

2x1 + 4x2 5x3 = 4.

pi.(x2 =

32x3, x1 = 12x3, x3 = x3

), (x2 = 32x3 1, x3 = x3, x1 = 12x3 + 4),

.

5.3.5. 5.3.1 - 5.3.4 pi.


1.

1 2 33 0 21 2 0

x1x2x3

= 12

3

pi. x = 10917

1823

.2.

1 2 33 0 21 2 0

x1x2x3

= 32

1

pi. x = 297

1823

.3.

5 2 13 0 22 2 1

x1x2x3

= 32

1

pi. x = t17

4t1 + 1

32t1 + 1

.

.

5. 88

4.

5 2 0 03 2 0 00 0 1 30 0 2 4

x1x2x3x4

=

3212

pi. x =

12141525

.

5.

5 2 0 03 2 1 00 0 1 30 0 2 4

x1x2x3x4

=

3212

pi. x =

35

01525

..

6.

5 2 0 00 2 1 00 0 1 30 0 0 4

x1x2x3x4

=

3212

pi. x =

310341212

.5.3.7. 5.3.6 pi.

5.3.8. pi .

1.

1 a ba 1 aa+ 1 a+ 1 a+ b

x1x2x3

= 11

2

pi. x = 1+bt1+aa

2 t11+a2

a+abt1+1at11+a2t1

.

2.

a b cb c aa+ b b+ c c+ a

x1x2x3

= 10

1

pi. x = cc

2 t1+abt1b2ac

b+bct1a2 t1b2act1

.

3.

1 2 33 2 11 1 1

x1x2x3

= 11

1/2

pi. x = t12t1 + 12

t1

.5.3.9. pi .

1.

1 a ba 1 ab a 1

x1x2x3

= 11

1

pi. x = a1b2a2+12a+b+1

b2a2+1 a1b2a2+1

.

2.

a b cb c ac b a

x1x2x3

= 10

0

pi. x = ac2+bcaba2 a

c2+bcaba2b+c

c2+bcaba2

.3.

1 a 00 1 a0 0 1

x1x2x3

= 00

1

pi. x = a2a1

.

.

5. 89

4.

1 a 0 ... 00 1 a ... 00 0 1 ... a... ... ... ... ...0 0 0 ... 1

x1x2x3...xN

=

000...3

pi. x =3aN1

...3a2

3a3

.5.3.10. pi a pi pi ; a 2 11 1 a

1 2 1

x1x2x3

= 00

0

pi. a = 1 a = 1

2.

5.3.11. pi a, b pi pi ; a 2 11 1 a1 2 1

x1x2x3

= 00

0

pi. a = 1 b , a 6= 1 b = 1 , a 6= 1 b = 1. .

5.3.12. pi pi a. a 2 11 1 a

1 2 1

x1x2x3

= 10

1

pi. a = 1 pi . a = 1

2 . a / {1, 1/2}

.

.

6pi Gauss

6.1

6.1.1. ( )

a11x1+a12x2+...+a1NxN = b1a21x1+a22x2+...+a2NxN = b2...aM1x1+aM2x2+...+aMNxN = bM

. (6.1)

6.1.2. pi pi

[A b

]=

a11 a12 ... a1N b1a21 a22 ... a2N b2.. .. ... ... ...aM1 aM2 ... aMN bN

.6.1.3. M N pi C pi n1,n2, ..., nM

1. 1 = n1 < n2 < ... < nM .

2. m = 1, 2, ...,M cmn = 0 n < nm cmnm 6= 0.

6.1.4. c11, c2n2 , ..., c2nM . n1 = 1, n2 = 2, n3 = 3, ... ,nM = M , pi .

6.1.5. pi Ax = b pi pi[

A b].

() , Ax = b () . pi - (. xn1, xn2, ... pipi) pipi.

90

.

6. GAUSS 91

6.1.6. pi pi pi pi M N pi

C =

r1r2...rM

. pi :

1. : rm rn, rm rn2. pipi pi : rm pi pi-

pi , rm krm, pi k 6= 03. pi : rm pi

pipi rn, rm rm + krn.6.1.7. pi , . - pi pi, pi, . pi :

1. rm rn rm rn2. rm krm rm 1krm3. rm rm + krn rm rm krn.

6.1.8. pi C pi pi C, C= EC, pi E pi pi pipi pi I pi. - pi E pi.

6.1.9. pi E pi pi - E = E1 pi pipi pi I pi.

C= EC C = E1C.6.1.10. Ax = b. pi pi C =

[A b

],

pi C =

[A b

]. Ax = b Ax = b, .

x pi pi pi pi .

6.1.11. pi Gauss pi Ax = b, :

1. pi pi pi C =[

A b]

C =

[A b

].

2. pi Ax = b, pi pi pi .

.

6. GAUSS 92

6.1.12. Ax = b. pi pi C =[

A b],

pi C =

[A b

]. Ax = b Ax = b, .

x pi pi pi pi .

6.1.13. Ax = b pi pi , pi pi .

1. pi 0 = bm 6= 0 ( ).

2. 0 = bm 6= 0 pi .

3. .

6.1.14. M N pi A Ax = b. C = [ A b ] pipi C =

[A b

] C.

K1 A K2 C. .

1. K1 < K2: ( ).

2. K1 = K2 < N : pi .

3. K1 = K2 = N : .

6.1.15. : Ax = 0 () Ax = b, b 6= 0 ( ). x .

1. x = x + u , u .

2. x pi x =x + u, pi u .

6.1.16. M N pi C pipi, pi pi , .

1. 1.

2. .

6.1.17. pi M N pi A pi pi.

6.1.18. N N pi A. pi .1. pi A1.

2. A = E1E2...EK , pi E1, E2, ...EK pi.

.

6. GAUSS 93

6.2

6.2.1. 2x1 + x2 + x3 = 1

4x1 + x2 = 22x1 + x2 + x3 = 7

.

pi. pi 2 pi pi 2 pi :

2x1 + x2 + x3 = 14x1 + x2 2 (2x1 + x2 + x3) = 2 2 1

2x1 + x2 + x3 + (2x1 + x2 + x3) = 7 + 1 1

2x1 + x2 + x3 = 1x2 2x3 = 42x2 + 2x3 = 8

.

pi pi

2x1 + x2 + x3 = 1x2 2x3 = 4

2x2 + 2x3 + 2 (x2 2x3) = 8 + 2 (4)

2x1 + x2 + x3 = 1x2 2x3 = 4

2x3 = 0.

, . ( ;). .

2x3 = 0 x3 = 0,x2 2x3 = 4 x2 = 4,

2x1 + x2 + x3 = 1 2x1 = 1 4 0 x1 = 32.

. ( ) x1 = 32 , x2 = 4, x3 = 0.6.2.2. pi 2 1 14 1 0

2 1 1

x1x2x3

= 12

7

. pi pi.

pi. pi pi 2 1 1 14 1 0 22 1 1 7

. (6.2)pi pi r2 r2 + (2) r1 r3 r3 + 1 r1 ( 2pi pi 2 4: 2 = 4

2 1 pi pi 2

2: 1 = 22 , pi ainxn pi

pi r3 r3 + k r1, pi k = ainajn ). (6.2) 2 1 1 14 + (2) 2 1 + (2) 1 0 + (2) 1 2 + (2) 12 + 1 2 1 + 1 1 1 + 1 1 7 + 1 1

= 2 1 1 10 1 2 4

0 2 2 8

(6.3)

.

6. GAUSS 94

pi r3 r3 + 2 r2 (6.3) 2 1 1 10 1 2 40 + 2 0 2 + 2 (1) 2 + 2 (2) 8 + 2 (4)

= 2 1 1 10 1 2 4

0 0 2 0

(6.4)pi pi pi (6.4) pi pi

A =

2 1 10 1 20 0 2

, b = 14

0

pi 2 1 10 1 2

0 0 2

x1x2x3

= 14

0

.

2x1 + x2 + x3 = 1

x2 2x3 = 42x3 = 0

pi pi pi--pi-:

2x3 = 0 x3 = 0,x2 2x3 = 4 x2 2 0 = 4 x2 = 4,

2x1 + x2 + x3 = 1 2x1 = 1 4 0 x1 = 32.

. ( ) x1 = 32 , x2 = 4, x3 = 0.6.2.3. pi pi pi pi pi rm rn .

pi. pi E pipi pi m n pi. E : E1 = E. .. , pi r2 r4 pi

E =

1 0 0 00 0 0 10 0 1 00 1 0 0

E1=

1 0 0 00 0 0 10 0 1 00 1 0 0

.

.

6. GAUSS 95

pipi pi pipipi pi

x1 + x2 + x3 + x4 = 1

x3 + 2x4 = 1

x2 + x3 3x4 = 22x1 + 3x3 + x4 = 4.

6.2.4. pi pi pi pi pi rm krm(k 6= 0) .

pi. pi E pipi pi pipi m pi pi k , . m- E

[0 0 ... 0 k 0 ... 0] .

E1 pipi pi pipi m pipi 1/k , . m- E [

0 0 ... 01

k0 ... 0

].

.. , pi r2 3r2 pi

E =

1 0 0 00 3 0 00 0 1 00 0 0 1

E1=

1 0 0 00 1

30 0

0 0 1 00 0 0 1

. pipi pi pipipi pi

x1 + x2 + x3 + x4 = 1

x3 + 2x4 = 1

x2 + x3 3x4 = 22x1 + 3x3 + x4 = 4.

6.2.5. pi pi pi pi pi rm rm+krn .

pi. pi E pipi pi pi k m pi, . m- E

[0 0 ... 0 k 0 ... 0 1 0 ... 0] .

.

6. GAUSS 96

E1 pipi pi pi k m pi,. m- E

[0 0 ... 0 k 0 ... 0 1 0 ... 0] ... , pi r2 r2 + 5r1 pi

E =

1 0 0 05 1 0 00 0 1 00 0 0 1

E1=

1 0 0 05 1 0 00 0 1 00 0 0 1

. pipi pi pipipi pi

x1 + x2 + x3 + x4 = 1

5x1 + x2 + x3 3x4 = 2x3 + 2x4 = 1

2x1 + 3x3 + x4 = 4.

6.2.6. 6.2.1 pi 2 1 14 1 02 1 1

x1x2x3

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