= Solution
Write $G=M^{-1}N$. Subtracting the exact equation $Mx=Nx+b$ from the iteration gives $e_{\ell+1}=Ge_\ell$, where $e_\ell=x-x^{(\ell)}$. Hence $e_\ell=G^\ell e_0$. Also
$$
\det(N-\lambda M)=\det(M)\det(G-\lambda I),
$$
so the stated scalars are precisely the <eigenvalues> of the <iteration matrix> $G$.
The <spectral radius criterion for matrix powers> states that $G^\ell\to0$ if and only if all its <eigenvalues> have modulus strictly less than one. Applying that theorem gives $e_\ell\to0$ for every initial vector. Therefore $\boxed{x^{(\ell)}\to A^{-1}b}$.
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