Unformatted text preview: A i similar to B i for all i , prove that A 1 âŠ• Â·Â·Â· âŠ• A t is similar to B 1 âŠ• Â·Â·Â· âŠ• B t . Solution. Absent. 4.61 Prove that an n Ã— n matrix A with entries in a f eld k is singular if and only if 0 is an eigenvalue of A . Solution. Absent. 4.62 Let A be an n Ã— n matrix over a f eld k . If c is an eigenvalue of A , prove, for all m â‰¥ 1, that c m is an eigenvalue of A m . Solution. Absent. 4.63 Find all possible eigenvalues of n Ã— n matrices A over R for which A and A 2 are similar. Solution. Absent. 4.64 Prove that all the eigenvalues of a nilpotent matrix are 0. Use the CayleyHamilton theorem to prove the converse: if all the eigenvalues of a matrix A are 0, then A is nilpotent....
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 Fall '11
 KeithCornell

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