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1、LinearAlgebraFinalExam(A)2004-20051.Fillingintheblanks(3’×6=18’)(1)Letbe(4×4)matrices,anddet(A)=4,det(B)=1,thendet(A+B)=.(2)Let,then.(3)Letisandimensionvector,and,thenthematrixAB=.(4)LetAbea(3×3)matrix,and1,2,3aretheeigenvaluesofA.ThentheeigenvaluesofA*are.(5)Letbealinearlydependentsetofv

2、ectors,where.Thenthenumberkis.(6)Let.Thenthecrossproduct=.2.Determiningthefollowingstatementwhetheritistrue(T)orfalse(F)(2’×6=12’)3(1)IfAandBaresymmetric(n×n)matrices,thenABisalsosymmetric.()(2)IfAandBarenonsingular(n×n)matricessuchthatA2=IandB2=I,then,(AB)-1=BA()(3)Ifu·v=0,theneitheru=0o

3、rv=0.()(4)IfAisnonsingularwithA-1=AT,thendet(A)=1()(5)IfAandBarediagonal(n×n)matrices,thendet(A+B)=det(A)+det(B).()(6)IfAisan(n×n)matrixandcisascalar,thendet(cA)=cdet(A).()3.(15’)Calculatethedeterminantofthematrix.4.(15’)Considerthesystemofequations,determineconditionsonkthatarenecessarya

4、ndsufficientforthesystemtobehasonlysolution,infinitesolutions,andnosolution,andexpressthesolutionsbyvectors.35.(10’)LetbeasetofnonzerovectorsinRmsuchthatwhen.Showthatthesetislinearlyindependent.6.(15’)Letand.FindB.7.(15’)Letisaneigenvectorof,(Ⅰ)Findthenumbersaandb.(Ⅱ)Findtheeigenvaluesoft

5、hematrixA.3