# On the determinant of a sum: when does $\det(A+B)=\det(A)+\det(B)$ hold?

When does $\det(A+B)=\det(A)+\det(B)$ hold?

Is there necessary and sufficient condition?

• Related: Does det(A+B)=det(A)+det(B) hold?. – dxiv Oct 24 '16 at 22:36
• @dxiv may be but only one post there (by PolyaPal) is relevant. – T.... Oct 24 '16 at 22:36
• It is related (and I didn't say duplicate). One of the answers appears to provide a certain sufficient condition. – dxiv Oct 24 '16 at 22:40
• @dxiv not that is not relevant (only Polayapal's is relevant). – T.... Oct 24 '16 at 22:41
• An obvious case is: When the dimension is $1$ (that is, the matrices consist of just one number each). Another obvious case: If one of the two matrices is the zero matrix. A less obvious case: $A$ and $B$ are upper triangular matrices, and one of them has only zeros on the diagonal. I don't think those cases are exhaustive. – celtschk Oct 24 '16 at 22:53

It is a long comment to show that problem in arbitrary dimension is far from trivial. If we set simplified case: $B = I$, then: $$n = 2: \det(I + A) = 1 + \det(A) + Tr(A)$$$$n = 3: \det(I + A) = 1 + \det(A) + Tr(A) + \frac{Tr^2(A) - Tr(A^2)}{2}$$$$n = 4: \det(I + A) = 1 + \det(A) + Tr(A) + \frac{Tr^2(A) - Tr(A^2)}{2} + \frac{Tr^3(A) - 3Tr(A)Tr(A^2) + 2Tr(A^3)}{6}$$ General sequence comes from relation between determinant and exponent of the trace of the matrix logarithm. I doubt that there is a trivial solution in general case.