Matrices- Nilpotent

Quote:

Originally Posted by amm345

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

Since N is a nilpotent square matrix, $N^k = 0$ 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

$I - N + N^2 - N^3 + ,,,, + N^{k-1}$ .

To verify this,
$(I + N)(I - N + N^2 - N^3 + ,,,, + N^{k-1})$
$= I - N + N^2 + ,,,, + N^{k-1} +N - N^2 + ...- N^{k-1} + N^{k}$
$= I$ , since $N^k = 0$ .

If k is an even positive number, then the required inverse is
$I - N + N^2 - N^3 +,,,, - N^{k-1}$ . You can verify this in the same manner as the above.