# Thread: Determinants of Sums and Differences

1. ## Determinants of Sums and Differences

Hi,
I was wondering if there was a relation between
det(A+B) and det(A) and det(B)
where A and B are n by n matrices.

Also, I was wondering what the reation between
det(A-B) and det(A) and det(B).

Thanks for any help.
I don't know if a relation is supposed to exist by the way...

2. Look at these examples. Then you decide.
$\displaystyle \begin{array}{l} A = \left[ {\begin{array}{cc} 1 & 3 \\ 1 & 2 \\ \end{array}} \right]\quad \& \quad B = \left[ {\begin{array}{cc} 3 & 1 \\ 1 & 1 \\ \end{array}} \right] \\ A + B = \left[ {\begin{array}{cc} 4 & 4 \\ 2 & 3 \\ \end{array}} \right]\quad \& \quad A - B = \left[ {\begin{array}{cc} { - 2} & 2 \\ 0 & 1 \\ \end{array}} \right] \\ \left| A \right| = - 1,\quad \left| B \right| = 2,\quad \left| {A + B} \right| = 4,\quad \left| {A - B} \right| = - 2 \\ \end{array}$

3. Originally Posted by tbyou87
Hi,
I was wondering if there was a relation between
det(A+B) and det(A) and det(B)
where A and B are n by n matrices.

Also, I was wondering what the reation between
det(A-B) and det(A) and det(B).

Thanks for any help.
I don't know if a relation is supposed to exist by the way...
The determinant function for square matrices,
$\displaystyle \det: M_{nn}\to \mathbb{R}$
Is a homomorphism for product (not addition).
Thus,
$\displaystyle \det (AB)=\det(A)\det(B)$

This is mine 44th Post!!!

4. ## I think I get it

So... I don't see any pattern. Thus, if i'm not missing anything, then there is no relation between det(A), det(B) and det(A + B).

If you can comfirm this, thank you very much.

Also thanks for both of the previous replies.

5. Originally Posted by tbyou87
So... I don't see any pattern. Thus, if i'm not missing anything, then there is no relation between det(A), det(B) and det(A + B).

If you can comfirm this, thank you very much.

Also thanks for both of the previous replies.
No there is no relationship.
(Not any nice one).

6. Originally Posted by tbyou87
Hi,
I was wondering if there was a relation between
det(A+B) and det(A) and det(B)
where A and B are n by n matrices.

Also, I was wondering what the reation between
det(A-B) and det(A) and det(B).

Thanks for any help.
I don't know if a relation is supposed to exist by the way...
There is no such relationship. To demonstrate this one need only find
matrices A, A', B, B' such that det(A)=det(A'), and det(B)=det(B') and
det(A+B) != det(A'+B'). Which with a bit of trial and error is easy enough
to do.

This demonstrates that there is no function f, such that:

f(det(A),det(B))=det(A+B).

(This is a stonger statement than IPH's that there is no simple relationship
between the determinants of two matrices and the determinant of the sum
of the matrices).

RonL