1:

1

1. 1. 1.1 1. . m n , P

mxn P. ij , i -

, - ij.

Amxn = [aij]mxn , (i=1,...,m;j=1,...,n)

mxn. (A)ij i- j- A. :

3x2, B 3x3, C 3x1. 2: . , m=n n, n. B 3. mxn = [aij]mxn . m =1, A1xn =[ a12 a12 . . . . a1n] - n=1 ,

- .

=

mnmm

n

n

aaa

aaaaaa

A

..............

......

21

22221

11211

=

1

21

11

1

.

.

m

mx

a

aa

A

+=

=

=

ii

iCBA

2

1,

752021435

,52302/11

1:

2

n=

nnnn

n

n

aaa

aaaaaa

KKKKKKK

21

22221

11211

i=j , aij

i=1,,n, .

, , , .

3. . Amxn = [aij]mxn e . -Amxn = [-aij]mxn .

4. . , mxn Omxn.

5. . Dn = [dij] 0, dij=0 za ij, .

6. . C , d11=d22=. . . =dnn=c C

n.

=

c

cc

Cn

......0.........0...000...00

7. . nxn, 1 n En In E, I .

, 3,

=

100010001

3I .

8. . A = [aij]mxn. () A () A

=

nn

nxn

d

dd

D

0......0..........0...000...00

22

11

1:

3

AT A'. A'=[a'ij]nxm A=[aij]mxn a'ij = aji (AT)ij=(A)ji. :

9. . A = [aij]mxn B = [bij]pxq , m=p n=q aij=bij i=1,2,...,m; j=1,2,...,n. :

2x-y=1 x+y=2 x-y=0 x+3y=4. 1.1 , (AT)T = A . : Amxn=[aij]mxn ATnxm =[aij]nxm a'ij = aji. (AT)T=[aij] mxn, aij=aji=aij. 10. . n, Anxn aij=aji i,j=1,2,...,n; .

1.2 S n, , ST=S. 11. . = [aij] n. aij=-aji . 1.3 An T = -A.

=

=

241302

,210432 'AA

=

=

322212

312111

3231

2221

1211

aaaaaa

Aaaaaaa

A T

=

++

4021

32

yxyxyxyx

1:

4

1.2 . , (+) () : a,b,c 1. a+b, ab ();

2. a+(b+c)=(a+b)+c, a(bc)=(ab)c ( ); 3. a+b=b+a, ab=ba ( ); 3. , 0, (

) a, a+0=0+a=a;

4. a3 b a+b=b+a=0. ( .) a a;

5. , 1, ( ), a, a1=1a=a ;

6. a, a0, b, ab=ba=1. ( 0, .) a-1.

7. a,b,c, a(b+c)=ab+ac ( ).

, (+,), . , P + () (), 1-7 . , (+,) Q(+,), C(+,) . Z2={0,1}, 2, :

1:

5

+ 0 1 0 1 0 0 1 0 0 0 1 1 0 1 0 1

, . 12. . A=[aij] B=[bij] mxn, C=[cij] mxn cij A B cij = aij + bij (i=1,2, . . . ,m; j=1,2, . . . ,n) A + B = C

:

:

C A B. 13. . A=[aij] mxn 3 C=[cij] mxn

cij = aij (i=1,2, . . . ,m; j=1,2, . . . ,n) C = A.

:

(-1)A -A. 14. A B = A + (-1) B. A, B C ,3 3 :

+++

++++++

=

+

mnmnmmmm

nn

nn

mnmm

n

n

mnmm

n

n

bababa

babababababa

bbb

bbbbbb

aaa

aaaaaa

...............................................

......

........................

......

.........................

......

2211

2222222121

1112121111

21

22221

11211

21

22221

11211

=

=

=3112

,010321

,025432

CBA

=

++=+

015113

001205342312

BA

=

==

=

082016128

042454443424

4,4,025432

AA

1:

6

1.4 A + B = B + A () 1.5 (A + B) + C = A + (B + C) () 1.6 A + O = O + A = A ( O A)

1.7 A + (-A) = (-A) + A = O 1.8 (A + B) = A + B

1.9 ( + ) A = A + A 1.10 () A = (A) 1.11 1 A = A 1.12 (A + B)T = AT + BT 1.13 (A)T = AT 15. A B mxn nxp C mxp,

C= AB. ij i - A j- B. , . a11 a12 . . . a1n b11 . b1j . b1p . . . . . . b21 . b2j . b2p

ai1 ai2 . . . ain . . . . . = . . . . . . . . . . .

am1 am2 . . . amn . . . . . bn1 . bnj . bnp

===

===

===

n

kkpmk

n

kkjmk

n

kkmk

m

kkpik

m

kkjik

n

kkik

m

kkpk

m

kkjk

n

kkk

bababa

bababa

bababa

1111

1111

11

11

111

......

.................

......

..................

......

:

),...,2,1;,...2,1(1

pjmibacn

kkjikij ===

=

++++++=

232213212222122121221121

231213112212121121121111

232221

131211

2221

1211

babababababababababababa

bbbbbb

aaaa

1:

7

: 1.14 (AB)C = A(BC) () : A mxn, B nxp C pxq . AB mxp, (AB)C mxq, BC nxq, A(BC) mxq. .

ij

n

ssjis

n

s

p

kkjskis

p

k

n

skjskis

p

kkjikij

BCABCA

CBACBACABCAB

))(()(

)()()())((

1

1 11 11

==

===

=

= == ==

:

1.15(A + B)C = AC + BC A(B + C) = AB + AC () ! AB BA. , . 16. . AB = BA A B . AB BA , C=AB-BA A B. 1.16 A = [aij] mxn , a D =[dii] nxn C = AD jjijij dac = . . A = [aij] mxn , a D =[dii] mxm C = DA cij=diiaij. , . : A = [aij] mxn , a D =[dii] nxn

C = AD jjijn

kkjikij dadac ==

=1,

dkj=0 kj. A = [aij] mxn , a D =[dii] mxm , C = DA ijii

n

kkjikij adadc ==

=1,

dik=0 ik.

=

++++++=

=

=

15125672

023512250215033213220312

,010321

,2532

BABA

1:

8

1.17 A e mxn , a In Im , AIn = ImA = A.

Ap = A A ...A (p=1,2,. . . ). 0=I. p

1.18 Ap Aq = Ap+q

1.19 (AB)T = BT AT. : A mxn, B nxp, AB mxp, (AB)T pxn . BT pxn, AT nxm, BTAT pxm . ,

ijTT

n

kkj

Tik

Tn

kik

Tkj

Tn

kkijkjiij

T BAABBABAABAB )()()()()()()()())((111

===== ===

17. . A () B

AB=BA=I. . :

.

1.20 . : AB=BA=I AC=CA=I.

B=BI=B(AC)=(BA)C=IC=C.

B=C, A . , A-1. ,

=

=

5273

3275

,1001

3275

5273

,1001

5273

3275

imatricite

1:

9

21 (A-1)-1=A 22 A B , AB

(AB)-1=B-1A-1. : A B .

(AB)(B-1A-1)=A(BB-1)A-1=AA-1=I, (B-1A-1)(AB)=B-1(A-1A)B=B-1B=I.

2.2 2.2.1 mxn :

1. () . 2. ()

. 3. ()

(). 2.2.2 . B A A B A~B. 2.2.3 . In . Eij i- j- . Ei() i- . Eij() j- i- . Eij, Ei() Eij() , , , : Eij = Eij, Ei() = Ei() Eij() = Eji() 2.2.1 3:

=

=

=

100010201

200010001

100001010

)2(13)2(312 EEE

1:

10

2.2.2.

=

100001010

1E

. A B

E1A E1B. :

2.2.3

=

=

=

100010201

200010001

100001010

' )2(13)2(312 EEE

2.2.1 Eij, Ei() Eij(), Eij, Ei() Eij().

: A mxn, Im m. Eks k- s- . ,

ik,s eij=

=

jiji

,1,0 ekj=

=

sjsj

,1,0

i esj=

=

kjkj

,1,0

.

B=EksA, B, j{1,,n} :

ik,s bij= ijijiim

lljil aaeae ==

=1,

sjsjks

m

lljklkj aaeaeb ===

=1, kjkjsk

n

lljslsj aaeaeb ===

=1

B k- s- A. Ek() k- . ,

1:

11

ik, eij=

=

jiji

,1,0 ekj=

=

kjkj

,,0

. B=Ek() A, B j{1,,n} :

ik, bij= ijijiim

lljil aaeae ==

=1, kjkjkk

m

lljklkj aaeaeb ===

=1,

B k- A . Eks() s- k- . , ,

ik eij=

=

jiji

,1,0 ekj=

==

sjkj

skj

,,1

,,0

.

B=Eks() A, B j{1,,n} : ik, bij= ijijii

m

lljil aaeae ==

=1,

sjkjsjkskjkk

m

lljklkj aaaeaeaeb +=+==

=1,

B s- k- A. A nxn . 2.2.2 A B mxn , A ( )

B 1, 2, . . . , k B = EkEk-1E2E1A, ( B = A E1E2Ek).

: A B, B A k . . B = EkEk-1E2E1A 1, 2, . . . , k B = EkEk-1E2E1A B A . 2.2.3 ()

,

1:

12

(Eij)-1 = Eij, (Ei())-1 = Ei(1/) (Eij())-1= Eij(-). 2.2.4 A=[aij] mxn

() mxn

=

0..0...000........0..0...000

......000.........

......0......

222322

11131211

rnrr

nr

nr

bb

bbbbbbbbb

B bii0 i=1,. . . , r.

2.2.5 A=[aij] mxn mxn

)()()(

)(

rnxrmxrrm

rnrxr

OOOI

., Ir r. 2.2.6

, .

2.2.7

. 2.2.8 .

, , -1.

: A () . , Ek . . . E2 E1 A = I, Ei i{1, 2, . . . ,k} , -1 Ek . . . E2 E1 AA-1 = I A-1 Ek . . . E2 E1 I = A-1

A. , , , A.

1:

13

2.2.4: =

011111121

.

: , [A|I] :

-1=

110011

111. -1=I

.

1:

14

2.3 2.3.1 .

A=[aij] n, tr A,

, : 2.3.1 tr A = trA

, 2.3.2 tr AT = tr A.

A .

a11a22-a12a21

2.3.1

. A 3.

=

=n

iiiaAtr

1

=2221

1211

aaaa

A

103)2(414321

det,4321 ====

= AAA

=

333231

232221

131211

aaaaaaaaa

A

211222112221

1211det aaaaaaaa

AA ===

1:

15

. , , -.

, , , . , . . 123, , . 123, 132, 231, 213, 312 321. , , 123, 231 312 , , 132, 231 321, . . , . 2.3.2 . S n .

S S . , S={a,b,c}

S. S n 1,2,...,n,

=

=

abccba

gacbcba

f ,

.

det

312213322113332112

312312322311332211

333231

232221

131211

aaaaaaaaa

aaaaaaaaaaaaaaaaaa

A

+

+==

.332112322311312213

322113312312332211

3231

2221

1211

333231

232221

131211

aaaaaaaaa

aaaaaaaaaaaaa

aa

aaaaaaaaa

++=

1:

16

S, S 1,2,...,n. ,

, I= (i1, i2, i3, . . . , in), ik {1,2,3,...,n}. 1 n. 2.3.3 n

123n=n! 2.3.3 . i, j

, i j, i>j.

, 2413 2, 1 , 4 , 1 4, 3 . , ? . . 2 2, 2. .. . , 35142 2 , 3 ( ), 0 , 1 , 2+3+0+1=6. 2.3.4. .

, . I= (i1, i2, i3, . . . , in) Ip .

, 35142 , 6 , 12345 0 , , 21345 . 2.3.4 ,

, , , .

niiiin

......321

321

1:

17

, n. 2.3.5 .

n, A n , ( ). ,

+, (i1, i2, ..., in) -, .

I=(i1, i2, . . . ,in) {1,2,...,n}, a Ip . det A. , , n!, n!/2 n!/2 . , , , .

=

mnmm

n

n

aaa

aaaaaa

A

..............

......

21

22221

11211

nniiiaaa ...

21 21

=

==),...,,(

21

21

22221

11211

21

21...)1(

............

...

...

detn

n

p

iiiIniii

I

nnnn

n

n

aaa

aaa

aaaaaa

A

1:

18

2.3.5 det AT = det A.

: . , . 2.3.6 ( ) ,

. : , 1 ( ) . . .

A. (i1, i2, ..., ik, ..., ij, ..., in), . 1 (i1, i2, ..., ik, ..., ij, ..., in), ik ij, - - . , 1 . det A1 = -det A. 2.3.2:

2.3.7 ( )

.

: , , , . , det A = -detA, , det A=0. 2.3.8 - n- ,

, .. akj=bkj+ckj, (j=1,2,...,n), 1

njk nijikiiiaaaaa .........

21 21

362141051

141362051

=

1:

19

2 , - , 1 bk1, bk2, ..., bkn, 2 ck1, ck2, ..., ckn.

: ,

A1 A2. 2.3.9 ( )

, .

: A=[aij] 1 - .

2.3.10 ( ) .

: , . 2.3.11 D , det D = a11a22...ann.

2.3.12

, . : 2.3.8,

nknk

nkknk

nikiiinikiii

nikikiiinikiii

acaaabaa

acbaaaaaa

............

)......(......

2121

2121

2121

2121

+=+=

AaaaaaaaaAI

kikiiiI

Inikiii

Ink

p

nk

p det......)1(......)1(det2121 21211

===

1:

20

, . 2.4 2.4.1 , . aij,

i- j- n, Aij n-1 i- j- . ( n-1) aij Aij Mij=detAij. ( ) aij

a*ij = (-1)i+j det Aij= (-1)i+j Mij

2.4.1:

2.4.1 A ,

aij, , = ijij aaAdet .

A

aaa

aaa

aaa

aaaaaa

aaa

aaa

aaa

aaaaaa

aaa

aaa

aaa

aaaaaa

aaa

aaa

aaa

aaaaaa

aaa

aaa

aaaaaa

aaaaaa

nnnn

jnjj

jnjj

n

n

nnnn

jnjj

knkk

n

n

nnnn

jnjj

jnjj

n

n

nnnn

jnjj

knkk

n

n

nnnn

jnjj

jnknjkjk

n

n

det

...........

.............

............

...

...

...........

.............

............

...

...

...........

.............

............

...

...

...........

.............

............

...

...

......................

......................

...................

.....

.......

21

21

21

22221

11211

21

21

21

22221

11211

21

21

21

22221

11211

21

21

21

22221

11211

21

21

2211

22221

11211

=+

=+=+++

3331

2321

3331

23212112

3331

232112

333231

232221

131211

)1(,,aaaa

aaaa

aaaaa

Aaaaaaaaaa

A ==

=

= +

1:

21

: i=j=1.

1111)...(

211)...(

212

2

21

21...)1(...)1(det MaaaaaaaA

n

n

p

n

n

p

iinii

I

iiiniii

I === 11 . i- , j- , aij , i-1 . . , j- j-1 . . 1 ,

AAA jiji det)1(det)1(det 111++ == . 1 aij

(1,1) ,

++ === ijijijijjiji aaMaAA )1(det)1(det 1 .

2.4.2

( ) . ,

: 2.3.8, i- n-1 , aij=0+. . . +0 +aij+0. . . +0,

= ijij

nnnjnn

ij

nj

aa

aaaa

a

aaaa

..............

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

......

21

111211

i{1,2,. . . n}. ++= ininii aaaaA ....det 11 .

: aij A Aij n-1, ( i-) ( j-) .

= =

+

= =+

==

==n

j

n

jijij

jiijij

n

i

n

iijij

jiijij

MaaaA

iliMaaaA

1 1

1 1

)1(det

)1(det

1:

22

i- ( j- ) , . , . !

. aij aij* : 2.4.3

: ' i-

k- , 2.3.12, ' . ', i-

2.4.4 A B , det (AB) = det A det B. 2.4.2 . A*

(i,j) aij* aij A. (A*)T A.

2.4.5 A=[aij] n,

A(A*)T=(A*)TA=AIn

==+++

==+++

sjakosjakoA

aaaaaa

ikiakokiakoA

aaaaaa

nsnjsjsij

kninkiki

0det

...

0det

...

221

2211

.0

detdet)('det

1

1111

ikakoaa

AaaAaaaaaaaA

n

jijkj

n

jijkj

n

jijkj

n

jijij

n

jijkjij

=

=+=+=+=

=

=

=

=

=

1:

23

: (*).

bij=ai1aj1*+ai2aj2*+ . . . +ainajn* , 36 bii=A, bij=0 ij. A(A*)T A A(A*)T=AI, I A. (A*)TA=AI. 2.4.2:

, . 2.4.5 A A0.

: A0, . 2.4.5,

A(A*)T=(A*)TA=AIn

, A

A , A_1 AA-1=A-1A=I.

=

=

nnnn

iniji

nj

nnjnn

nj

nj

nnnn

inii

n

T

bbb

bbb

bbb

aaa

aaa

aaa

aaa

aaa

aaa

AA

............

............

......

............

......

......

........

........

...

*)(

21

1

1111

1

2212

1111

21

21

11211

IAAAA

AAAAko

TT

T

4400040004

315222404

212021

211*)(*)(

315222404

*)(,4,212021

211

=

=

==

==

=

IAAA

AA

A TT == )*)(1()*)(1(

TAA

A *)(11 =

1:

24

2.4.4 AA-1=I AA-1=1

A0. . k mxn. 2.4.3 k mxn

k- k- , kmin{n,m}.

2.4.4

r(A).

2.4.6 A n, A n. 2.4.7 mxn, r(A) min{n,m}. : 2.4.8 r(A-1)=r(A)=n. 2.4.9 r(AB) min{r(A),r(B)}.

2.4.10 r(ATA)=r(AAT)=r(A).