Why can't we just multiply by the inverse of K in the equation of the eigenvalue problem?

Question

What happens if we multiply by $K^{-1}$ from the left in $$A k = \lambda k $$? , a little back story: Was trying to find a simpler way to prove that $P(A)K=P(\lambda)K$ and then thought of doing what I asked about in the title until I arrived at $P(A)=P(\lambda)$ which is what I though of as the statement of Cayley-Hamilton, now I know that it should be only applied to the characteristic equation so I'm lost now...

Answer 1

First of all in $$AK=\lambda K$$ K is a vector not a matrix so we can not find $K^{-1}$ so we have to solve for the non-zero vector $K$ to satisfy $$AK=\lambda K$$

Secondly you mentioned that $$P(A)=P(\lambda)$$ and that is not the case because $P(A)$ is a matrix and $P(\lambda) $ is a polynomial. The Cayley- Hamilton theorem is about $P(A)=0$ for the characteristic polynomial of the matrix $A$ not the equality $P(A)=P(\lambda)$ which have different dimensions.