11 Answer only one of the following two alternatives.

EITHER

A \(3\times3\) matrix \(A\) has distinct eigenvalues \(2\), \(1\), \(3\), with corresponding eigenvectors \(\begin{pmatrix}1\\1\\0\end{pmatrix}\), \(\begin{pmatrix}-1\\0\\b\end{pmatrix}\), \(\begin{pmatrix}0\\1\\-1\end{pmatrix}\), respectively, where \(b\) is a positive constant.

(i) Find \(A\) in terms of \(b\).

(ii) Find \(A^{-1}\begin{pmatrix}0\\2\\-2\end{pmatrix}\).

(iii) It is given that \(A^n\begin{pmatrix}1\\1\\0\end{pmatrix}=\begin{pmatrix}4\\4\\0\end{pmatrix}\) and \(A^n\begin{pmatrix}-1\\0\\b\end{pmatrix}=\begin{pmatrix}-1\\0\\b^{-1}\end{pmatrix}\). Find the values of \(n\) and \(b\).