Question

Prove that if M is nilpotent, then det(M)=0

Answer 1

can it be equivalently said that if inverse of matrix exist then it's determinant exist??

Answer 2

if the inverse of a matrix exists, then det(A) is not zero

Answer 3

hmmm

Answer 4

maybe it has to do with that if at some point det(A^m)=0, then det(A)=0 because you can't multiply by anything else to get 0

Answer 5

http://en.wikipedia.org/wiki/Nilpotent_matrix

Let M be non zero matrix.
M^k = M x M^(k-1) = 0

let M' (if it exist) be inverse of M,
M'M x M^(k-1) = 0

Inductively you can prove that M = 0, which means M' is not inverse of M, so There is no inverse of M.

Answer 6

thanks

Answer 7

seriously ,,, i don't know anything about this subject.

Answer 8

no? it's advanced linear algebra....I used the fact that det(AB)=det(A)det(B)

Answer 9

oh ... thats great.