Linear Algebra with Applications (3rd Edition)

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Unformatted text preview: Math ass. Spring sum 1 Exam# 1 l. Totem-False“? T F Fore'li'et'jpr invertible an matrixA there existsanonzere nxn man'ix Bsuehthat AH isthezerornatrix. E —k - T F Thematr‘tx L2 is invertible for all real numbers k. T F Iftwo invertible rnataieesA and B comrntne. then A" and 3" must commute as well. - T F The fonnnla dettA+Hl =dettA}+det{B) holes forafl 2x2 matrices A 31113. T F If A isa 33:3 matrix such thattjrt+tjjl1 =0. then-1A mustbe invertible 2. 1n parts {a} through tel . 5ch a matrix of the required form. or state that no such matrix exists. I 2 a. An invertible 2x2mattixrt such that A" =[ l 3 4 bah Exlmett‘ixaltttith ketiAl=spanlfl I c. lit 2x2 matrixAsuch that im{A]:lm(A2). a. a 2:3 meflilfisuehthet kertAJ={fi}. e. A nonzero 2x 2 matrixfll such that lilfl[2A) = Zdetljrlj. 3. Consider the manioes A through E below. 13.36 —flI.4E 41.48 0.154 as as 1 3:3 or c: as —o.s o 3 43.3 0.6 E _ l l} ens —o.s' "' -t l Fill in the blanks in tlte sentences below. We are told that there is a solution in eaeh ease. Matrix _ represents a dilation Matrix represems a projection Matrix represents a shear Matrix represents a reflection Matrix represents a rotation [1—1 5. Show that the rotation matrix can be written as the product of three shear matrices (you can use horizontal as well as vertical shears). Math 253, Spring 2W1, Exam 1, Solutions .a. F If AfizflJhen multiplywitlt A" fi'onttltelefito seetltat 3:0. In. F Thedetermjntntia3+k’,aothattitomat:ixFailltobeinvetfibleihrk=—2. 1:. 'I‘ lfAE=BA,then {AB}‘1={EA}" andtlnta 3"A'1=A"B’l. d. F If'lrelet A=E=I:,thmdet[.ll+B}=4hutdetiA]+¢et{B}=2. e T A‘+2A+I,=o.mhai —A’—1£=I,.A{—A—2i,]=i’i.and A'L = ‘A_H: _ ..|"'_‘__1 " —2_ —2 l. 1"" A't’! } _ 31-3 tl’Ltz —H2 —3 2 —3o 2:: 1" A'La 2 -:o to mommamdbmarbiWyoonmtntkaatoocofwhiehiamm. e. A=[: :1].forexample,widi in1{.«{}=qian[:lland in1(,4=]={ii]. , fiarezample. More generally, 3113' matrix ofthe form A =[ a ThereisnosuchmatrifoAiaa2x3matri1,tltenthesyatmfi=fihaa infiniter mart}.r solutions. since them is at least one free vari able. e. Anjrnonzerononintrertihlernatrixliwill do,si:noethen detflA}=2det{A}=fl. _ I k t} 3. Brepreaentsadtlanon.offl1efm'm|fl It} I: l toad —ai;nfl sin 9 coat? diagonal and opposite entries oil" the diagonal. C represents a pmjectirm. with parallel column vectors, and A! represents. a reflection (by the prooeas of elimination]. Note that: the eohamn vectors offit are perpendicular unit 1I.rr.:t.'.tora, as required ofa reflection matrix. D—l_la1fllcl+rtfia+c+nbcmm iii—leltil a 1+be' “3H: it: it; Tl- Uaing an mflognus appmaoh, we find the altelnatitre representation If :11] = . (These two are the only solutions] 1 it E repreaents a {vertical} Ihear, of the Form l , D rcpt-meats a rotation, of the form l,withtwo identiealenntesonthe 4. Two approaches: Write ...
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This homework help was uploaded on 01/23/2008 for the course MATH 253 taught by Professor Ghitza during the Spring '07 term at Colby.

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exam1spring2001 - Math ass Spring sum 1 Exam 1 l...

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