Linear algebra

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Homework Statement

a) find all the 2x2 matrices where AA=A.
b) prove that if
$$I + a_{1}A + a_{2}A^2 + ... + a_{k}A^k = 0$$
then A is invertable

Homework Equations

1)det(A) = 0 iff A isn't invertable

The Attempt at a Solution

a) I'm not sure how to approch this. I found that if A is invertable then the only solution is A=I but how do i cover the other cases?

b) by rearanging:
$$A(a_{1}I + a_{2}A^1 + ... + a_{k}A^{k-1}) = -I$$
and if i take the determinant on each side i see that |A| <> 0 so it's invertable. Is that correct?
Thanks.

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b) looks correct.

Regarding a), assume A is regular and see what follows from that. Further on, consider A = I as a special case.

Dick