Hi, If A is an nxn matrix then the ranks of A^n and A^(n+1) are equal. I need help to prove this assertion. Thank's in advance.
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can't all be distinct since these are integers between 0 and n so for some ,
Last edited by Idea; Feb 27th 2017 at 11:54 AM.
Yes,but maybe from now on it start to increase and ker(a^n) is strictly contained in ker(A^(n+1))?
It is not true that rank$(A^n)$=rank$(A^{n+1})$ for every non-negative integer n. Idea has proved for you that there is some j with rank$(A^j)$=rank$(A^{j+1})$
We can prove that Let Use this and induction to show that