Given a square matrix M of dimension n : In order to find the set of eigenvalues of M one should solve the equation : $$\\M.X=\lambda.X \enspace \enspace \enspace be \enspace \enspace \enspace (M-\lambda.I_{n}).X=0 \\ where \enspace \lambda \enspace is \enspace an \enspace eigenvalue \enspace of \enspace M \enspace and \enspace X \enspace is \enspace a \enspace non-nul \enspace column \enspace vector \enspace called \enspace the \enspace eigenvector \enspace of \enspace M \enspace relative \enspace to \enspace the \enspace eigenvalue \enspace \lambda$$
The french wikipedia article states that since X is non-null, this means the matrix (M-λ.I) should be a singular matrix that is its deterimnent is equal to zero. My question is : What does the singularity of some matrix has to do with the nullability of its product with another vector ?