
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN: 9781305658004
Author: Ron Larson
Publisher: Cengage Learning
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Transcribed Image Text:The Cayley-Hamilton theorem from Linear Algebra states that if A is an n x n matrix
with characteristic polynomial p(λ-det(A-AI) then p(A) 0, where the last 0 is
the n × n zero matrix. Verify that Cayley-Hamilton holds for the matrix
2 -2 0
A 0 2 0
-2 2 2
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Suppose that A is an invertible matrix over and O is a zero matrix. Prove that if AX=O, then X=O.
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a Find a symmetric matrix B such that B2=A for A=[2112] b Generalize the result of part a by proving that if A is an nn symmetric matrix with positive eigenvalues, then there exists a symmetric matrix B such that B2=A.
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