SOLUTION: an n *n matrix A is called nilpotent if, for some positive integer k, A^k = o, where o is the n*n zero matrix. Prove that id A is nilpotent, the det A = 0.

SOLUTION: an n n matrix A is called nilpotent if, for some positive integer k, A^k = o, where o is the nn zero matrix. Prove that id A is nilpotent, the det A = 0.

Question 1200894: an n *n matrix A is called nilpotent if, for some positive integer k, A^k = o, where o is the n*n zero matrix. Prove that id A is nilpotent, the det A = 0.
Answer by ikleyn(52818) (Show Source): You can put this solution on YOUR website!
.


For any matrix A,   = .


If A is a nilpotent matrix, then   = 0;  hence   = 0.


It implies that for nilpotent matrix A   = 0  for some integer k > 0;  hence,  det(A) = 0.    QED

Solved.

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