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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.
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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,
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“114).; EWUQJ A \s (c) If A and B are invertible matrices in MnanF), then det(ABA‘1B“1) = 1.
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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
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I 567‘ “($519575 f Vaclav: jerxotu 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—).
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 Fall '08
 GUREVITCH
 Linear Algebra, Algebra, Matrices, Invertible matrix, following incorrect statement, Mam Macaw

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