exam3sol - 1(10 points each Determine whether each of the...

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Unformatted text preview: 1. (10 points each) Determine whether each of the following assertions is true or false. Give a brief explanation for each answer (full proof is not required). (a) If A E Mnxn(]R) and n is odd, then A has an eigenvector. Tam; y<>>=erHA—I\1) :; a f0£7ww¢€ 4 W “n W #2. w S‘WLQ m is 09W, 10%) {Ms 0:! £14571 mu rawh/ 34) A W5 04— (amt me le’ul 6M0? wMS/aoxwa’vj agar/earn, (b) If A is an n x n matrix with two distinct eigenvalues A1 and A2, and dim(E,\1) = n — 1, then A is diagonalizable. T95 3M0: A2 ‘0 cm et‘DrowQuz, fi‘m (511321. . Wee wMCEAfll‘oQa‘MiEM) 27!- We (ill/MATS W s W, 5;, ovatemwamraa w. 9:7 1m mum 7w wagw‘sabxul} “114).; EWUQJ A \s (c) If A and B are invertible matrices in MnanF), then det(ABA‘1B“1) = 1. Tag on (A6 A"e“ ) : old (A) 004+ («(5) am (A") 062+ Us") : cannula “calm ohiw zi 2. (20 points) Give a correct version of the following incorrect statement: Given a matrix A E menOF) and a column vector 1) E F”, there is a vector 1100 E IF” such that the solution set of the system of equations szb consists of the vectors at = $0 + v for all m the nullspace‘ N(A). V ., a. EMW , an ft, is forest“, I 567‘ “($519575 f Vaclav: jerxot-u w“ «fie/Wu. . 3. (20 points) Express the last entrylxit of the solution of the system 4221 :01 1 33'00'x2_0 2441m3“0 3301s4 0 in terms of determinants. Leave the determinants in your answer unevaluated. You may assume that the coefficient matrix of this system is invertible. ' ' ‘ qall 422—) = det 3 5‘90 3 30c> fl? 1‘*# 0 &d 2.4qt 5300 3304 4. (30 points) Calculate A2009, where - 2—). 12...,90 '.‘ ~13 2“)(‘>=s)-o Mag , Mam Macaw “H '12 v , , x —( §)/:‘Z)~o -> )6,”sz , v.,=(§) will» lug“de ’ ULM C2: 11,5) I 19 O (’0 = 06-3%.?" =A ‘ (Li: $300k :4 U want «cw musxcwj \Lv C“{C°~(9—L€ Q“ 4» mu m WWW w AM :A] ...
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