\(A=\begin{pmatrix}a&3\\4&a+4\end{pmatrix}.\)

(i) Find the values of the constant \(a\) for which \(A^{-1}\) does not exist.

(ii) Given that \(a=4\), find \(A^{-1}\).

(iii) Hence find the matrix \(B\) such that \(AB=\begin{pmatrix}2&3\\4&-5\end{pmatrix}\).