Adv Alegbra HW Solutions 139

Adv Alegbra HW Solutions 139 - A i similar to B i for all i...

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139 (iv) Prove that every hermitian matrix A over C is diagonalizable. Solution. Absent. 4.55 If A is an m × n matrix over a f eld k , prove that rank ( A ) d if and only if A has a nonsingular d × d submatrix. Conclude that rank ( A ) is the maximum such d . Solution. Absent. 4.56 (i) If A and B are n × n matrices with entries in a commutative ring R , prove that tr ( AB ) = tr ( BA ) . Solution. Absent. (ii) Using part (i) of this exercise, give another proof of Corollary 4.96: if A and B are similar matrices with entries in a f eld k , then tr ( A ) = tr ( B ) . Solution. Absent. 4.57 If A is an n × n matrix over a f eld k , where n 2, prove that det ( adj ( A )) = det ( A ) n 1 . Solution. Absent. 4.58 If C = C ( g ) is the companion matrix of g ( x ) k [ x ] , prove that the characteristic polynomial h C ( x ) = det ( xI C ) = g ( x ) . Solution. Absent. 4.59 Let R be a commutative ring. If A 1 ,..., A t are square matrices over R , prove that det ( A 1 ⊕···⊕ A t ) = det ( A 1 ) ··· det ( A t ). Solution. Absent. 4.60 If A 1 ,..., A t and B 1 ,..., B t are square matrices with
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Unformatted text preview: A i similar to B i for all i , prove that A 1 ⊕ ··· ⊕ A t is similar to B 1 ⊕ ··· ⊕ B t . Solution. Absent. 4.61 Prove that an n × 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 × n matrix over a f eld k . If c is an eigenvalue of A , prove, for all m ≥ 1, that c m is an eigenvalue of A m . Solution. Absent. 4.63 Find all possible eigenvalues of n × 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 Cayley-Hamilton theorem to prove the converse: if all the eigenvalues of a matrix A are 0, then A is nilpotent....
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