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