A question in finding a invert of a certain operator

[transgalactic]

http://img371.imageshack.us/my.php?image=38787787nq3.jpg

i don't know how to act
i only have some formula

there is no matrix to work on
what do i need to do??

 

[HallsofIvy]

You are given that A²- A+ I= 0. Just basic algebra tells you that is the same as I= A- A²= A(I- A). Now, stop worrying about formulas, etc. and think about what A^-1 means.

 

[transgalactic]

it means that if we take the image of a certain basis
and put it into the A^-1 we get the original basis

how its going to help me??

thought of certain solution of making some basic multiplication
operations

i got the resolt

A^-1=I-A
is it ok??

 

[HallsofIvy]

it means that if we take the image of a certain basis
and put it into the A^-1 we get the original basis

?? No, A^-1 doesn't mean anything like that: it means the linear operation that "undoes" A. A^-1Ax= x and AA^-1x= x.

how its going to help me??

thought of certain solution of making some basic multiplication
operations

i got the resolt

A^-1=I-A
is it ok??

My point was that A(I- A)= I shows that I -A does exactly what we want of the inverse: AA^-1= I. Since A(I-A)= I, (and inverses are unique), A^-1= I- A.

 

[transgalactic]

so what is the answer??
what is the value of A^-1