2J_midterm_soln

2J_midterm_soln - Math U MidtermExam A ru"r",K F...

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Unformatted text preview: Math U MidtermExam A ru"r",K F Y Date: LOl29/07 Thistest consists 5 questions of worth 40 points each.Readall directionscarefully.Showall work neededto arrive at yoursolutions. Cleorly indicateand labelyourfinol answers. Have fun! 1) A.) Determine whethereachof the following your response. statements uue or false. are Circle A andB areassumedto n x n matrices x is an n x 1 vector. be and 1" pf" each A,lo,.r, a. A homogeneous linearsystem haveinfinitely i-of"l can manysolutions. False product two elementary b. The of matrices an elementary is matrix. True r An overdetermined system always manysolutions. l4J$nitely @ rrue (f"fg d. det(/ + B) = det(.A)+ det (B) rrue e. f. A*"8) --_-___ False g. Thedeterminant a matrix thesln_qlitseigenvalues. of is True (f"/r"-) h, A matrix is diagonalizable only if all the eiffnvaluesare distinct. i. tfAandare B ."* j. then,ffif;il"*inantsequat. are "J,iili,ent GAtse\ True 9"") Ax=o,th;;-&-(a) Folse = 0. lf x is a nonzero vejlgrq{ B.)Forthe matrices given below, possible if compute each the following. of ?l s e"..r' % t "t 6: 1 0 p f5 Ll a . A+ B ,B ^=[iil= |itl r, ,=[\3] Mct pr"iblu b . z c - B .fo /l l3 ql c. eB:Is 9l d. CA [z tl lY 5 l ble A/dfpos-', e' o' I 1- '[_ " 'l tl 'l - -3 1 x1- 2x2 2l A.)Consider system linear the givenby of equations lz rt -4 x 2 * 5 x 3 = -6 ( x r - 2 x 2 * 5 x 3- -3 a. Rewrite system linear the of equations a matrix as equation. to ,{s rl /i l?1^fl/ Larts ^l=l_ll f r -? olf b. Solve system linear the of equations using Gaussian Elimination. 11. I I -2 O '-3 1 -Le , r Q . 1 I r - T lO O li^-o l t -6 1 r-,- Lt'15 3J lr /\' - R z+ Rsr Rs LO -1 s,,-tJ 6 ' ,O ) o: - 31 t 5; O l -(,vRr+R, I t -L O i-31 O f o o 5 ' ,o I { R,> Q ,,-r -z o [ i_31 ,', Xr=O X .= 4 X r"-3rl o1, ^r l;3 ;,iJ x= f-'j^1 lo of J = B.)Given matrix, to pls =f 1 0 'o-'l], ; [j I lJ A-'^=[ x= i ? t 0 0l L- o jl=l{l :ll to Ax -1 | andA = l2l, findthe solution the equation = b. 1J L3I 0I l1 l 3) A.)compute determinant matrix = the ofthe , - )t' [j j j] /i3s/ l l l ?l = tt:lt rl':l /351 l'sl ? (s ' ,.xr))- (s- e) =m : ^l-v (No B.)Using matrix A from above,find eachof the following the values. workrequired be shown.) to -- -'l a. l,r!^','o\i,o det(.47) '! b. det(A-1) = -l T c. det(AA-1) = I d li i 2l=-g' hssl = e ll 1 gl ,l lr rzl 4l A.)Writea specific example a 3 x 3 matrixin row echelon form,but not in reduced echelon of row form. D Yt5 E*) L. /o o r l f r 131 l o | 1l B.)Writea specific matrix example a system linear of of in formwhichis inconsistent. equations augmented EOLt o, IA p+t o o: 3l C.)Write a specific exampleof a systemof lineareguationsin augmentedmatrixform which has infinitelymany solutions. l0 p15 r') U sit) ft D.)Prove that similar matrices have same the determinant. l 0 pts P,ool' a^n) 6 are 6 iniLcnr ,f 5 iauer*iale ' f3, S-'A S $or same {'f't'on de+($)' d,el(s -'45) b7 s'-'bs = AL+( r)olol(n\ JrtG) s' (s-t) = d.el(w)Jetk) obt ' dul&) 'Iet(s J ') = slelCk) J*G) bl, o$ in verseby de(i^ifrr:n con'na{nli'e h'y f*P'"ft b) bl. c)et|cfr -del-Cq) 't'+Cs) B\=Jet6fi Azt($ 'I"LW uVf/' #J 7 5) A.i Find eigenvalues eigenvectors the matrix = | . ?1, the e for and L -4 1 l' )i -. ' i: i B.i Findmatrices and X whichdiagonalize matrixA above. D the r1 "-t ^_f X=l tt i [.i Math U Midterm ExamB t/ rv Name: Ft E I Date: L0/29/07 This test consists 5 questions of worth 40 pointseach. Reod directions oll carefully. Show work neededto arrive at olt your solutions.clearly indicateond labetyour Havefun! final onswers. 1) A.) Determine whethereachof the following statements true or false. are yourresponse. Circle Aand B areassumedto n x n matrices x is ann x 1 vector. be and a. A homogeneous linear system cannot hayeinfinitely manysolutions. rrue b. An overdetermined system always [email protected] @g rrue manysolutions. e. A triangular matrixp-singular True 6 d. det(d+ B) = det(.A) det(B) + X True \LaJg c. Theproductof two elementary [email protected] an elementary matrix. @!tP if andonly if the diagonal entries all nonzero. are E. A matrix is diagonalizable only if all of its e-igenvalues distinct. are t. rfl, isan ",r"nu.,rffien Vyp 1/), irF.onlt"tie.nuarue ofA-1. Folse rrue h. lf A and B are row equivalent then theirdterminants equal. are <fulsO rrue i. j. lf x is a nonzero veggrlg Ax=O, then det(24): 6. Thedeterminant a matrixisthe sumof its eigenvalues. of e9J9 @r. Fatse [1 =[ 21 ' B=E ' t=[| i] il a . A+ B AlotpossiUle "1 1: B.) Forthe matrices givenbelow, if possible computeeachof the following. b . c _ 2 8 = l_t _ ql L - / - tJ CA Not Po#iula AB= I -I L8 4r > [-r AT I L L7 5 I/s 7l ;l ,'il A.)consider system tinear given the of equations by {r2I ( t , * x 2 * Z x r= 3 a. Rewrite system linearequations a matrixequation. the of as =fil l ? i il/il [t o 3) -zft,tl.-+R, lt r., *i:::: b. Solve system linear the of Elimination. equations using Gaussian f I o z O T. L l tr I c ' 3 ..{ C l ^ .r sJ [r [o IO lo 7 o o/ I L 3l 31 -k,+esae3 t O L | nf O 3 c) t o o =* Rr'Rrn *r /\- , f r O 2 3 o l ., o jtl iil ' aaJ X.2o( Yz = O Xt ' 3'Lo\ L0 o c) ,. Is or X =l o z 'tl I /.t J tQ,nR- o z .\_-, [r I o lo rl B ',,.*= and find )Givenp ; il,*'= [_1, +1 b=[i], the solutionto the equationAx = b, ; x'A',o'[j,ifl[il=[i,] 3) A.)Compute determinant matrix = the ofthe , [l , 2l 5J l_1 4 l,,4sl ti;/ t i i ?l =ll,r/ I / 2. '1, , = (1zrcs) - (" - *) -cr)ro) : 2- 3 =f/i B.)Using matrix A from above, find eachof the followingvalues. the (No workrequiredto be shown.) a. det(A-l) = -l b. det ( / " ) = c. det(.4/-1) = -l I | d lii2 l =-7 h 4 sl l " t1i ;;l1 = I 22 1 1 1 0l 4l A.) Writea specific example a 3 x 3 matrixin row echelon of form, but not in reduced echelon row form. A See vLrsion B.)Writea specific example a system linearequations augmented in of of matrixform whichis inconsistent. C.)Write a specific in matrixform whichhasinfinitelymany example a system linearequations augmented of of solutions. D.)Prove that similarmatrices have samedeterminant. the 5) A.)Find eigenvalues eigenvectors the t.tri* A = [? the and for T] = d"*in'tr) / t, I I l-1 1 1I (-a)(') : (s-r)tr-r)=(A-q)(x = o '4 \= ' 1 ,1 : nt .^ +t i'-6) : x='l uur;.=!o r-fl =';). ,t:" =+ !^,, f'*^, f Ax-Ax \'L' l:.', f ?rt';,':;'=;:,=) /=lrl B.) FindmatricesD and X which diagonalize matrixA above. the 1 ,=1 e J "- Lo o I "=f t ? ll Math 2JMidterm ExamC Name:KEI Date: LOl29l07 all worth 40 pointseach.Read directionscarefully.Showall work neededto arrive at of Thistest consists 5 questions Hove your solutions. indicdteand labelyourfinal answers. Clearly fun! your response' Circle are statements true or false. whethereachof the following 1) A.) Determine and to A and B areassumed be n x n matrices x is an n x L vector' has\finitelymanysolutions. alwaVs system a. An underdetermined True matrix. is of b. Theproduct two elenentarymatrices not an elementary Fatse @ manysolutions. infinitely can linearsystem heJve c. A homogeneous Fatse @ d. det(.A+ B) = det(/) + det (B) .-\ @@' True tEglp are entries all nonzero. if matrixis nonsingularandonlyif the diagonal e. A triangular f. of lf l\ is an eigenvalue g![then Vtr is an eigenvalue .A-1. Cr@ AlD Fotse Fatse @' are g. lf A andB arerowequivalent thenthelldeterminants equal. True Cr@ i. j. vectoratd Ax=O, then det(A) = I' h. lf x is a nonzero Fatse of Thedeterminant a matrixis the sgp-elits eigenvalues. rrue if A matrixis diagonalizableandonlA9f @!9 are its eigenvalues distinct. rrue @g eachof the following. givenbelow,if possible compute B.)Forthe matrices a . B+A Not posstbleb sc_ B= [,] ^=l), ,=lril il :] c=lril c. CA Nof .,, ,oossrbl d*=r; il It e . O'= [,, , ? ' ,J ,1 = -5 h -x z - 2x2 * Zxs = -L0 givenby of the 2l A.) Consider system linearequations l'*:I r -x z * 2 x , = -5 t equation' as of the a. Rewrite system linearequations a matrix ( 'lfl Ii,lillil o i-rJ -A ,tRo1R, -(,+Rsnftt Elimination' Gaussian using of the b. Solve system linearequations t -t-{ Ll -r Lt"sJ | . -2. L t-tctl /\-, -2R , 4, t R1 [ t z ol -sI L r - l 2,. lo o -l o -sl )R"-te" Lo o o , '. X,' O {r t d'( lo It -( o o I t ; o o ifl : lo 2. tt'd o lo o L Lo o L o jfl fl xr = -5+o( 0 0l or .= "] [-f [1 0 t.,r,# B.)Given =li] X=A'b=1, ii][i] r rl,A-t=lL [ -r o [l o r J Ax = h' to tn. L it l .no , = lt l, Rno solution theequation rJ t3l ol l2 l 3l A.) compute determinant matrix = the orthe , [i i i] | |il=l;!)-'l;tl ' | : (,'1 ,';- l) - ( t - , ) = =Fl 6-rl B.)Using matrix A from above, find eachof the followingvalues. (No workrequiredto be shown.) the = a. det(d-l) b. det(Ar)= c. det(ea-r; = 1 2 2 0 l| + Z | d. lr z Ll= h 4 sl Ll 11 101 e. li 4 il = -7 h zil 4l A') Write a specific example a 3 x 3 matrixin row echelon of form, but not in reduced echelon row form. lee A vcrsion B.)Write a specific example a system linearequations augmented of of in matrixform whichhasinfinitelymany solutions. C.)Writea specific example a system linearequations augmented of of in matrixform whichis inconsistent. D.)Prove that similarmatrices havethe samedeterminant. . r{ for and the 5) A.)Find eigenvalues eigenvectors the matrix = [ tj r), det(A-r -z =(t-e)(t-,)) - (Do lr-^ t lrlal =(^-3xa-do Ax,Ax = A,-SA+6 W. , {r r , (X r '::,":::-=) I -l -;:::' f:,' A-3, L \"L,(rr , - \ x z : 2x r - ) ( ^ = | :l x,+Y'"LF. ( ^ * ,-7 x ^ = 6 LtJ xt-\z=o ( the D B.)Findmatrices andX whichdiagonalize matrixA above. ' o=l3"gl 2 (l 10 "=[T ,ll J [f ...
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This note was uploaded on 04/07/2008 for the course MATH 44370 taught by Professor Sarahfrey during the Winter '08 term at UC Irvine.

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