[]1.1 1.2 nn1.3 1.4 1.5

[]1n2n3[]

()

--

65.

1. 2.3!,,,,.

(),,,

D

1DD',D=D' ()=D

2(),tk

ktD1=ktD1

(),ktD==0kt

3()==

1(),

2(),kt=0

4() =+=

5()k(),k+=43,

20=208

2(-3)84(-30/58)

nn-1aijij,n-1aij,Mij

DaijMij,Aij=(-1)i+jMijaijDa23,

()nD()

nD()

()(),(),(-3)(-3)4

Cramer (1),

k,12k0k9D0k=0k=9D=0

[]3.1 3.2 nn3.3 3.4 3.5

[]12345[]

??

Fmnaij (i=1,2,..,m; j=1,2,,n)Fmn,mn,aijij

(n,m),,

,(1).;()(2).;()(3)..()()(1).();()(2).();()(3).()().()

()

dr+10,()dr+1=0,r=n, ()*rn!

,dr+1=0 r=n, ,x1,x2,,xn,*

,dr+1=0 r

,:xr+1=c1,,xn=cn-r,, c1,,cn-r

x4=c,(c

,r=n,;r

1 2

3

4

n n

?

n,

nFna1,a2,,anniaii;n,,n1nnn1

,1nn1.,,(),n,-

()n,n,,+ - = + (- )()kF,kk,k

();();();();();,,n,on,klFFn,,,Fn

:

1, 2, , s sk1, k2, , ks , 1, 2, , s1, 2, , sx11 +x22 ++xss = :n(0,0,,0)n1, 2, , sk1= k2= = ks =0,

n1=(1,0,,0), 2=(0,1,,0),,n=(0,0,,1),n()n =(a1,a2,,an),n1, 2,, n,1, 2,, ni(1in)i=01+02+ + 1i+ +0n (1in)

=(2,3,-1), 1=(1,-1,2), 2=(-1,2,-3),3=(2,-3,5),1,2,3,

1, 2, , s,k1, k2, , ks ,k11+k22+ +kss=o1, 2, , s1, 2, , s,k1== ks=0,k11+k22++kss=o/x11 +x22 ++xss =o/o, 1,2, ,s,k0

1 2

1

2 1

A B

C D

( )

A B

C D

j1, j2, , jr: (1) j1, j2, , jr; (2) j1, j2, , jr.j1, j2, , jrj1, j2, , jr: (1) j1, j2, , jr; (2) , j1, j2, , jr, .

1,21,2,31,21,21,2,32,3 1,31234,

1031,2, ,s: r(1,2, ,s)1, 21,2, ,sr(1,2, ,s)=s11,()

()()()()()():A3.A:A. A

12(),()

13(),()

()()

14A=(aij)mn,A,A,A,A, A,a110A(),AA()r,A()r

A=(aij)mn,AA,Ar(A)

r(Amn)Amn().,0r(Amn) min{m,n}.r(Amn)=m,Amnr(Amn)=n,Amn()(),1(,)

;.:A,:

,:1,2,3

1

,

,

17:,r(A)=r().()1r(A)=r()=n.2r(A)=r()

11, 2,1+2.2,c, c1,,s,c11+c22++css

A B

C D

A B

C D

1,2,,s,1,2,,s.1. ,;2.r(A)=n,, .

18r(A)

,xr+1,,xn ,n-rn-rn-r:, 1,2,,n-r()

,,,1,2,,n-r., 1,2,,n-r()()

1,2,,n-r ,(): c1,c2,,cn-r()

1,, +21, 2,1-2190,1,2,,n-r,():= 0+c11 +c22 ++cn-rn-r ;ci0

D A B C D

mn A B C D

2

A B C D

,

,

_________________________.

A B

C D

1

[] 3.1 3.2 3.3 2.4 2.5 2.6

[]12345[]

mnA=(aij)mnB=(bij)mn,C=(aij+bij)mn(A,B),CAB;C=A+BA=(aij)mnB=(bij)mn,, aij=bij,AB,: A=B ,;

A-AA=(aij)mn,-A=(-aij)mn,AAmn-Bmn=Amn+(-Bmn)()A+BA-B.:

(1) A+B=B+A(2) (A+B)+C=A+(B+C)(3) A+O=O+A=A(4) A-A=A+(-A)=OA,B,CO

A=(aij)mn , kF. kAkA,;kA=(kaij)mn k(A+B)=kA+kB (k+l)A=kA+lA (kl)A=k(lA)=l(kA) 1A=AA,Bmn, k,lF

X,3A+2X=3B 3A+2X=3B 2X=3B - 3A

A=(aij)ms ,B=(bij)sn, ABmnC=(cij)mn; C=AB1AB, 2AB, A, B

CijcijAi Bj.

: BA

,AX=BAX=O, Om1

,AB,BA.22-16-32816220000=O1,ABBA,2AB=OA=OB=O

nn, n,nn()1,0n,n;En .E(I)

(1) (AB)C=A(BC);(2) A(B+C)=AB+AC;(3) (B+C)A=BA+CA;(4) Fk, k(AB)=(kA)B=A(kB);(5) Amn , Em Amn=Amn; AmnEn=AmnABAB=BA,AB,A, B

A.:AAX=XA::a,b.

An, k , AkAk=AAAk,l, ::(1) , (2) ,

:A,A,A'AT ATijaij'=Ajiaij

nnA,A|A|n: |A|=|AT|; |kA|=kn|A|; |AB|=|A||B| |A1A2As|=|A1||A2||As|.ABBA,|AB|=|A||B|=|B||A|=|BA|

An,|A|0,A();|A|=0,A()nABA2+AB+B2=O,B,A,A+BBAA+B

,

n,n:.;k;.

n,n()()A=aEn (An)Bmn, (aEm)Bmn=aBmn ; Bmn(aEn)= aBmn .,

n, n, (): (). k. .

nAAT=A,A.A=(aij)nn,aji=aij (i,j=1,2n).: AB, A+B. k.., .

nAAT=-A,A.A=(aij)nn,aji= -aij (i,j=1,2n).., .: (). k..

nA,BAB=BA=E,A, A;BA,A-1,A-1=B1. BA2. ABA-1=B,B-1=A3. A(BCA,AB=BA=E,AC=CA=E B=BE=B(AC)=(BA)C=EC=C)En., En En = En,(En)-1= En

AijnA|A|aij,A

A*AA* = A*A=|A|E000000

nA|A|0,A,AA

ABABn, AB =E,AB, A-1=B , B-1=A:AB =E, AB.AB =E:AB =E::BAAB,()

nA2A(A-E)=A3,E-A,(E-A)-1= A2 -A+E .

1.A,A-1,A*,(A-1)-1= A,(A*)-1=A/|A|.A-1A= E,A-1, (A-1)-1= A;A*A=|A|E,A*(A/|A|)=E.2.A,B, AB,(AB)-1=B-1A-1 .(AB)(B-1A-1)= A(BB-1)A-1= AEA-1 =E,(AB)-1=B-1A-1 .

3.A, AT,(AT)-1=(A-1)T .AT(A-1)T= (A-1A)T=ET=E,(AT)-1=(A-1)T .4.A, 5.An, 6.Ak0,kA,(kA)-1=k-1A-1 .

,

E(),(),E[i,j]E[i(k)]E[i,j(k)]

Eij(ij), ijij

kEi().ii

Ejki(Eikj)jkik

Amn, A,mAA,nA ,,A,,A,()

A, A-1.,G1,G2,,Gk,:A----n2n(A E)(A E), A,EA-1 .

AX=BAX=B(A).,X=A-1BA-1,A-1BAX=B (A)----n2n(A B)(A B), A,EA-1B .

X,AX+B=XAX+B=XB=X-AX=(E-A)XE-A: XA=B,A,X

A

A

-44

ABn

A

BABAB

C

D

B

A B

C D

X=

=

X=

X

X

[]4.1

[][]

AFn, F,n, A= ,A, AA, A, , .A= (o) - A = o(o)(E-A)=o (o)(E-A)X=o,

,1,2A,k,k, k11+k22(k1, k2)A

A5,=(1,1)TA5

nA2

C

A B

C6 D

n2

D

A B C D

n

A2

A5 B5

C6 D6

A(E-A)X=o(E-A)X=o|E-A|=0;|E-A|=0 (E-A)X=o;

|E-A|=0 A;(E-A)X=o,A

A.1= -2,2=1=-2(E-A)X=o:

A=-2k,k=1 ()(E-A)X=o:A=1= k11+ k22 ,k1, k2