sample_final_1_answ

# sample_final_1_answ - NOTE These are only answers to the...

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NOTE: These are only answers to the problems and not full solutions! On the Fnal exam you will be expected to show all steps used to obtain your answer. 1. Short Answer Problems a) det A =3 - ( - 2) = 5 so A - 1 = 1 5 ± 3 - 2 11 ² . b) L 1 = - 100 0 1 0 0 0 1 ,L 2 = 1 0 0 0 1 0 00 - 1 , so L 2 L 1 = - 1 0 0 010 - 1 c) Since C is 5 × 5 the characteristic polynomial of C will be a degree 5 polynomial and hence must have at least one real root. Thus, it has at least one real eigenvalue which will give a corresponding real eigenvector. d) Let ± b be in the columnspace of A . Then there exist an ±x R n such that A±x = ± b . But then A ± b = A ( A±x )= A 2 =0 = ± 0 . Hence, ± b is in the nullspace of A . 2. a) We row-reduce A to get 10 01 b)±rom our row-operations in a) we have A = 1 0 0 - 510 0 0 1 201 060 001 . 3. Since C is square and the nullspace is a subset of R 4 , C must be 4 × 4. Since the nullspace of A

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## This note was uploaded on 04/30/2010 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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sample_final_1_answ - NOTE These are only answers to the...

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