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Math Help - Determinants of Sums and Differences

  1. #1
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    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...
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  2. #2
    MHF Contributor

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    Look at these examples. Then you decide.
    <br />
\begin{array}{l}<br />
 A = \left[ {\begin{array}{cc}<br />
   1 & 3  \\<br />
   1 & 2  \\<br />
\end{array}} \right]\quad \& \quad B = \left[ {\begin{array}{cc}<br />
   3 & 1  \\<br />
   1 & 1  \\<br />
\end{array}} \right] \\ <br />
 A + B = \left[ {\begin{array}{cc}<br />
   4 & 4  \\<br />
   2 & 3  \\<br />
\end{array}} \right]\quad \& \quad A - B = \left[ {\begin{array}{cc}<br />
   { - 2} & 2  \\<br />
   0 & 1  \\<br />
\end{array}} \right] \\ <br />
 \left| A \right| =  - 1,\quad \left| B \right| = 2,\quad \left| {A + B} \right| = 4,\quad \left| {A - B} \right| =  - 2 \\ <br />
 \end{array}
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  3. #3
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    Quote Originally Posted by tbyou87 View Post
    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,
    \det: M_{nn}\to \mathbb{R}
    Is a homomorphism for product (not addition).
    Thus,
    \det (AB)=\det(A)\det(B)

    This is mine 44th Post!!!
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  4. #4
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    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.
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  5. #5
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    Quote Originally Posted by tbyou87 View Post
    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).
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by tbyou87 View Post
    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
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