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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