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March 3rd, 2012, 05:49 AM   #1
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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
frankpupu is offline  
March 3rd, 2012, 07:37 AM   #2
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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.
icemanfan is offline  
March 3rd, 2012, 12:55 PM   #3
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Re: matrix invertible problem

how do i delete this post?
mrtamborineman10 is offline  
March 6th, 2012, 11:45 AM   #4
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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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