In dimension $2$, we have the identity
$$\det(A+B) + \mbox{tr}(AB) = \det(A) + \det(B) + \mbox{tr}(A) \mbox{tr}(B)$$
but this is false when the dimension is $3$ or higher. So, we need that when dimension is $3$:
$$\det(A+B+C)=?$$
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A,B,C are just diagonal matrices. – Erhan Güler Apr 08 '21 at 18:57
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1See here for links with $n=3$. – Dietrich Burde Apr 08 '21 at 19:02
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4@ErhanGüler If they are diagonal, why didn't you include such important information in the question itself? – Rodrigo de Azevedo Apr 09 '21 at 10:39