Common Matrix Factor

Algebra Level 4

\[AB = BA'\]

Let \(A = \begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix} \), \(A' = \begin{bmatrix} 18 & 12 \\ -20 & -13 \end{bmatrix} \), and \(B = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), where \(a, b, c, d\) are positive integers, satisfying the equation above.

If \(\det(B) = 1\), compute \(a+b+c+d\).

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