show that if N is a nilpotent nxn matrix then identity matix+ N is invertible.

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- Sep 22nd 2009, 02:49 PMamm345Matrices- Nilpotent
show that if N is a nilpotent nxn matrix then identity matix+ N is invertible.

- Sep 22nd 2009, 05:42 PMaliceinwonderland
Since N is a nilpotent square matrix, for some positive integer k.

If k is an odd positive number greater than 1 (if k=1, then it is trivial), then the required inverse is

.

To verify this,

, since .

If k is an even positive number, then the required inverse is

. You can verify this in the same manner as the above.