Proving Invertibility of a Nilpotent Matrix: Formula for (I-B)^-1

[bonzy87]

1. If B is any nilpotent matrix prove that I - B is invertible and find a formula for (I-B)^-1 in terms of powers of B?



2. can't seem to figure out how to get the answer to this one, drawn a mental blank any help would be appreciated

 

[ircdan]

so B^k = 0 for some k

say B^2 = 0,

Notice (I-B)(I + B) = I

Now what if B^3 = 0

then (I-B)*something = I


generalize