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March 3rd, 2012, 05:49 AM  #1 
Member Joined: Jan 2012 Posts: 63 Thanks: 0  matrix invertible problem
Let A be n*n matrix such that A^k=0(n,n) (the n*n zero matrix) for some natural integer k. show that In+A is invertible. attempt solutions: for A^k=0(n,n) then we have A=0 or A is a matrix not equal to 0 ,but for some k s.t. A^k=0 for case 1, if A=0 then In+A =In ,and det(In+A) not zero then it is invertible for case 2. if A is not 0 then i dont know how to argue this one.cause for some A is not zero , maybe det(In+A)=0 or maybe A is not zero but A^k is not zero as well can someone give me some helps 
March 3rd, 2012, 07:37 AM  #2 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: matrix invertible problem
Not sure if this works, but I believe if then A must be of the form (or the transpose), in which case . Now, if you add to this matrix, you get , the determinant of which is , hence it is invertible. 
March 3rd, 2012, 12:55 PM  #3 
Member Joined: Jul 2010 Posts: 44 Thanks: 0  Re: matrix invertible problem
how do i delete this post?

March 6th, 2012, 11:45 AM  #4 
Newbie Joined: Dec 2011 Posts: 15 Thanks: 0  Re: matrix invertible problem
Suppose for some vector . So . This means that is an eigenvalue of A which is a contradiction since zero is the only eigenvalue of A.


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