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Proof of det of a matrix with submatrices

  1. Apr 11, 2012 #1
    1. The problem statement, all variables and given/known data
    PROBLEM 1:
    Let A,B,C,D four matrices nxn which are submatrices of matrix 2nx2n:
    [tex]E = \left( {\begin{array}{*{20}{c}}
    A&B\\
    C&D
    \end{array}} \right)[/tex]

    PROBLEM 2:

    Let A a matrix nxn, B a matrix nxm and C a matrix mxm. O is the null matrix mxn.

    2. Relevant equations
    PROOF 1:
    Say whether it's false or true that:
    [tex]\det (E) = \det (A)\det (D) - \det (B)\det (C)[/tex]

    PROOF 2:
    Proof that:
    [tex]\det \left( {\begin{array}{*{20}{c}}
    A&B\\
    O&C
    \end{array}} \right) = \det (A)\det (C)[/tex]

    3. The attempt at a solution
    No idea how to start. I know this is true for numbers but how can I make it generic for matrix? Just an idea?

    Thanks!!
     
  2. jcsd
  3. Apr 11, 2012 #2

    Dick

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    Science Advisor
    Homework Helper

    What's your definition of det?
     
  4. Apr 11, 2012 #3
    Well, we defined it first by taking the column numbers:

    [tex]|A| = {( - 1)^{1 + 1}}{a_{11}}\left| {{A_{11}}} \right| + ... + {( - 1)^{i + 1}}{a_{i1}}\left| {{A_{i1}}} \right| + ... + {( - 1)^{n + 1}}{a_{n1}}\left| {{A_{n1}}} \right|[/tex]

    also for rows..

    Thanks!

    BTW, what is this for? This is not a definition for submatrices but numbers.
     
    Last edited: Apr 11, 2012
  5. Apr 12, 2012 #4
    No idea?
     
  6. Apr 12, 2012 #5

    HallsofIvy

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    Staff Emeritus
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    That's not a definition until you tell what [itex]|A_{ij}|[/itex] means! And, you cannot define them as determinants in a definition of "determinants".

    Do you believe it is not necessary to know what a "determinant" is to prove something about a determinants?
     
  7. Apr 12, 2012 #6
    So, what's the definition? That's the one I have on my book. Just copied it.

    Thanks!

    PD: We defined the determinant of a matrix 2x2 as ad-bc and then we defined a 3x3 matrix and so on...
     
  8. Apr 12, 2012 #7

    Dick

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    Homework Helper

    I was hoping you had a definition more like http://en.wikipedia.org/wiki/Determinant in the "nxn matrices" section. You sum the products of elements from every row and column times a permutation factor. If you apply that to proof 2 you can see that none of the nonzero contribution to the determinant comes from B. So you may as well put B=0 as well. Now you can reduce the matrix to upper triangular without mixing A and C.

    It should be easy enough to find a counterexample for Proof 1.
     
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