exam1spring2001

Linear Algebra with Applications (3rd Edition)

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Page1 / 3

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online