Can you factorize the matrix?

Algebra Level 4

\[A^2 - 16A - 17I = 0_{2,2}\]

Let \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), where \(a,b,c,d\) are positive integers arranged in ascending order, and precisely two of \(a,b,c,d\) are prime numbers. These four numbers are also pairwise coprime.

With \(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \); \(0_{2,2} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \), satisfying the equation above.

If \(B = \begin{bmatrix} b & a \\ c & d \end{bmatrix} \), find \(\det(B)\).

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