Question

Let A be an n x n matrix. Show that if A^(k+1) = O, then I - A is nonsingular and (I - A)^-1 = I + A + A^2 + . . . + A^k

Answer 1

I think I have figured it out...

Answer 2

show me please

Answer 3

I was taught that if A is an n x n matrix, then 1/(I-A) = I + A +A^2 + A^3+....
but not relate to A^(k+1) =0

Answer 4

(I - A)(I + A + A^1 + A^2 + . . . + A^k)=

I(I + A + A^1 + A^2 + . . . + A^k) - A(I + A + A^1 + A^2 + . . . + A^k)

Answer 5

= (I + A + A^1 + A^2 + . . . + A^k) - (A + A^2 + A^3 . . . + A^(k + 1))

Answer 6

= (I - A^(k + 1)) = I - O , since A^(k+1) = O

Answer 7

= I

Thus, I - A is nonsingular and its inverse is equal to (I + A + A^1 + A^2 + . . . + A^k)

Answer 8

oh yea, you are brilliant!!

Answer 9

You know, I haven't done any matrix division yet. I spent like and hour and a half thinking about this and the moment I posted a question something just clicked haha.

Answer 10

Yes, I can see it. :) Congrat!! I wish I have 10% of your brain

Answer 11

Thanks for posting. I learn a lot

Answer 12

Thanks man. I see you interact a lot on here. Its great to see another person genuinely interested in mathematics.

Answer 13

:)