Matrices and Determinants!

Algebra Level 5

\[\large{S= \dfrac{\displaystyle \sum_{k=1}^{7} \det(A + \omega^{k-1}B) + \sum_{k=1}^{7} \det(B + \omega^{k-1}A) }{\det(A) + \det(B)} }\]

Let \(A,B \in M_{7} (\mathbb C)\) where \(M_{7} (\mathbb C)\) denotes a square matrix of order \(7 \times 7\) having complex entities in it. Let \(\large{\omega = e^{2\pi i / 7}}\). Then find the value of \(S\) upto three correct places of decimals.

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