Steady Determinant!

Algebra Level 4

\[\large{A^2 + B^2 + C^2 = AB+BC+CA}\]

Let \(n\) be a positive integer which is not a multiple of 3, and let \(A,B,C\) be \(n \times n\) matrices with real entries that satisfies the above equation. Find the value of the following correct upto three places of decimals:

\[\large{ \text{det} \Bigg((AB-BA)+(BC-CB)+(CA-AC) \Bigg) = \ ?}\]

Note: \(\text{det}(M)\) means the determinant of matrix \(M\).


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