Step in Euler's rotation theorem

Question

I have been examining the matrix proof for Euler's rotation theorem on Wikipedia. I have deduced every step up to proving that $\det (R - I) = 0$ for any rotation matrix R. However, I'm having difficulty understanding how to prove that $\det (R - I) = 0$ entails $(R - I)n = 0$. Any help in understanding this step would be greatly appreciated.

Answer 1

The determinant is zero implies one of the eigenvalues is zero implies there is a eigenvector to the eigenvalue zero implies the eigenvector to 0 obeys $(R-I)n=\lambda*n=0*n=0$ and therefore $Rn=n$