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## Homework Statement

(a) Show matrix A+I

_{n}is invertible and find (A+I

_{n})

^{-1}in terms of A.

(b) exp(A) =I

_{n}+A +(1/2!)A

^{2}+ (1/3!)A

^{3}+..+(1/2010!)A

^{2010}

Show exp(A) is invertible and find (exp(A))

^{-1}in terms of A

## Homework Equations

A is square matrix with size n*n such that A

^{2011}=0

## The Attempt at a Solution

From A

^{2011}=0

I made it to A*(A

^{2010})=0

it's in the same form as Ax=0, so can I say that solution, x=A

^{2010}

(well, I don't know whether this will help me so I just wrote it down)

(a) To show matrix (A+I

_{n}) is invertible, I tried to let (A+I

_{n})x=0

so

Ax+x=0

and I don't know how to continue anymore T^T

I just know if I can find out that solution for (A+I

_{n})x=0 is trival then it is invertible.

But, am I approaching it in a correct path?

(b) I guess we use the same approach as a?