Suppose thatp(A)= det(A-AI) has n distinct roots A,, . . .,A,. Prove directly that p(A)=
Chapter 3, Problem 14(choose chapter or problem)
Suppose thatp(A)= det(A-AI) has n distinct roots A,, . . .,A,. Prove directly that p(A)= (- l)"(A-AII). . . (A-AnI)=O. Hint: Write any vector x in the form x= xlvl + . . . + x,V" where vl, . . . , v" are n independent eigenvectors of A with eigenvalues A,,. . .,A, respectively, and conclude that p(A)x=O for all vectors x.
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