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.

Algebra ->  College  -> Linear Algebra -> 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.      Log On

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