Combinatorics involved in matrix

Let matrix \(A= \begin{bmatrix} x & y & -z \\ 1 & 2 & 3 \\ 1 & 1 & 2 \end{bmatrix}\)

where \(x,y,z \in \mathbb{N}\). If \(|adj(adj(adj(adj.A)))|=4^8.5^{16}\), then find the no. of positive integral solutions.


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