Relationship between determinant and trace of a matrix

Asked Jul 01 '23 at 00:50

Active Jul 01 '23 at 06:17

Viewed 137 times

Prove that for all matrices $A, B \in M_n(\mathbb C)$

$$ \det(A+B) = \det(A) + \det(B)+\operatorname{Tr}(A)\operatorname{Tr}(B) - \operatorname{Tr}(AB) $$

I have verified it by taking $n=2$ that is for $2\times 2$ matrices. Now how to solve it In general way?

edited Jul 01 '23 at 06:17

Rodrigo de Azevedo

asked Jul 01 '23 at 00:50

SUJAN DAS

But Tr(AB)=Tr(Identity matrix)=2 – SUJAN DAS Jul 01 '23 at 01:23
So it will be 1+1+4-2=4 – SUJAN DAS Jul 01 '23 at 01:24
This problem is taken from Titu Andreescu book – SUJAN DAS Jul 01 '23 at 01:30
The exercise in that book only says to verify it for $n=2$. I'm not sure that this holds for higher $n$, and would be very surprised if it did. – Charlie Jul 01 '23 at 01:32
3

@SUJANDAS Try it with $A=B=I_n$ for $n \gt 2$. See also Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$). – dxiv Jul 01 '23 at 01:32
Yeah got it. For n=4 the relation is not true. – SUJAN DAS Jul 01 '23 at 01:41
1

Consider the case where $A$ and $B$ are diagonal matrices, say $A= \text{diag}(a_1,...,a_n)$, $B = \text{diag}(b_1,...,b_n)$. Then $\text{det}(A+B) =\prod_{i=1}^n(a_i+b_i)$, while the right hand side is $\prod_{i=1}^n a_i + \prod_{i=1}^n b_i + (\sum_{i=1}^n a_i)(\sum_{j=1}^n b_j)$. The left-hand side is thus homogeneous of degree $n$, while the right-hand side has terms of degree $n$ and degree $2$, so this can never hold for $n \neq 2$. – krm2233 Jul 01 '23 at 04:45
Related – Rodrigo de Azevedo Jul 01 '23 at 06:16
@SUJANDAS If you are satisfied that the relation does not hold true for $n \gt 2$ then you should either post a self-answer to that effect, or otherwise retract the question altogether. – dxiv Jul 01 '23 at 06:24
@SUJANDAS refer to one of my questions, https://math.stackexchange.com/questions/4688492/let-a-and-b-are-square-matrices-of-order-2-with-real-elements-such-that-ab-a – NadiKeUssPar Jul 04 '23 at 12:11

Relationship between determinant and trace of a matrix

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