矩阵习题课(一).pdf

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1、ÝÝÝSSSKKK()SNJ;.~K~©ÛZuoLiancuiHomePageCollegeofMathematicalScience,IKTianjinNormalUniversityJJIIJIMay18,201111
25£¶w«'4òÑSNJMainContents;.~K~©ÛHomePage1.SNJIKJJII2.;.~K~©ÛJI12
25£¶w«'4òÑ1SNJ(1)Ý${:^£éA1ê!ê¤{5K£éA¤.(ÜÆ,Ƥá.SNJêþ¦{:kA=(ka)1ª

2、ê¦kD«;.~K~©ÛijmnO:n3

3、kAnn

4、=k

5、A

6、;

7、2A3×3

8、=2

9、A

10、;HomePage

11、−2A

12、=−8

13、A

14、.IK3×3JJIIk(A+B)=kA+kB,(k+l)A=kA+lA,JI13
25(kl)A=k(lA),1·A=A.£¶w«kE¡êþݽXþݶ'4(kE)A=A(kE)=kA,òÑ=ê¦Ý?Ó?¦{.¦{µ¦^:AêuB1êAB¦.¦{5K:A=(aik)mn,B=(bkj)ns,KXnAB=(cij)ms,cij=aikbkj.SNJ;.~K~©Ûk=1

15、(ÜƤáµ(AB)C=A(BC).HomePagem©ƵA(B+C)=AB+AC,IK(B+C)D=BD+CD.JJIIJIÆÚÆؤá,ddû½a14
25SK.£=:AT=òA1C!C1¤Ý¶w«.'4(A)T=(AT),AT(i,j)−=A(j,i)−mssmòÑ.TTTTT(A)=A,(A+B)=A+B,TTTTT(kA)=kA,(AB)=BA,−1TT−1(A)=(A),TTTT(A+A+···+A)=A+A+···+A,SNJ12s12s;.~K~©ÛTTTT(A

16、1A2···As)=As···A2A1.!a11a12···a1nHomePageAT=A,A=a..12a..22···a2..n¡é¡;IK...a1na2n···annJJII0a···aJI121n−a0···a15
25AT=−A,A=..12..2..n...£−a1n−a2n···0¶w«é¡.'4òÑ(2)ݦÈ1ª

17、AB

18、=

19、A

20、·

21、B

22、.(AB)≤min{A,B},P,Qþ_Ý,(PAQ)=(PA)=(AQ)=A.SNJ;.~K~©Û(3)_Ý:éAnn,e3Bnn

23、,¦AB=BA=E,K¡A_,B¡A_Ý.HomePageA_⇔

24、A

25、6=0.IK⇔Aòz(ÛÉ).JJII⇔A=n.JI16
25−11∗A_,A=A.£

26、A

27、¶w«_¡÷.'4òÑ1)A_,Ù_.2)A_⇒A−1_,AA−1p_Ý,=−1−1(A)=A.3)A,Bþn_,AB½_,(AB)−1=−1−1SNJBA.;.~K~©Û4)A_,AT½_,(AT)−1=(A−1)T.5)eAnn,Bnnþn,÷vAB=E,KA,BþHomePage_B=A−1.IKéuÝØäkd(Ø:XJJI

28、I100JI1000A=0100,B=010,17
250010110000£¶w«AB=E,A,Bþ,,Ø_.'46)k6=0A_,(kA)−1=k−1A−1.nnòÑ2;.~K~©Û(1)'uÝ$.SNJaabab···ab;.~K~©Û111121na2
a2b1a2b2···a2bn~1...b1b2···bn=.........,aabab···abHomePagenn1n2nna1IK
a2Pnbb···b=ab,JJII12n

29、...iii=1JIan18
25
a11a12b1x
a11xa12yb1xy1a12a22b2y=xy1a12xa22yb2£b1b2c1b1xb2yc¶w«=x(a11x+a12y+b1)+y(a12x+a22y+b2)+(b1x+b2y+c)22'4=a11x+2a12xy+a22y+2b1x+2b2y+còÑncosϕ−sinϕ~2O.sinϕcosϕcosϕ−sinϕ)A=,Ksinϕcosϕ2cosϕ−sinϕcosϕ−sinϕA=sinϕ

30、cosϕsinϕcosϕcos2ϕ−sin2ϕ−2sinϕcosϕSNJ=;.~K~©Û2cosϕsinϕcos2ϕ−sin2ϕcos2ϕ−sin2ϕ=.sin2ϕcos2ϕcos2ϕ−sin2ϕcosϕ−sinϕHomePageA3=A2A=sin2ϕcos2ϕsinϕcosϕIKcos2ϕcosϕ−sin2ϕsinϕ−cos2ϕsinϕ−sin2ϕcosϕ=sin2ϕcosϕ+cos2ϕsinϕ−si