Posted by David on Tuesday, February 9, 2010 at 5:37pm.
Suppose A is a square matrix satisfying the equation A^3  2I = 0. Prove that A and (A  I) respectively are invertible. (the hint is to find an explicit equation for A^1.
To prove A is invertible, this is what I did although I don't that it's right:
A^3  2I = 0
A(A^2  2A^1) = 0 because A * A^1 = I
so (A^2  2A^1) must = A^1
(A^2  2A^1) = A^1
so (1/3)A^2 = A^1 thus proving A is invertible. However I don't know that this is correct, and I don't know how to prove that (A  I) is also invertible. I know I need to factor the equation, but the 2I makes it difficult.

College Math : Linear Algebra  David, Thursday, February 11, 2010 at 5:50pm
Well I guess no one can help. :/
Oh well, I'll figure it out