Proof Of Crammer's Rule For Determinents

Discussion in 'Math' started by Glenn Holland, Aug 9, 2015.

  1. Glenn Holland

    Thread Starter Member

    Dec 26, 2014
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    I'm looking for the proof of Cramer's Rule for evaluating determinents or matrixes.

    Algebra books and sites give examples of how the rule is applied for solving simultaneous equations, but I would like to know the actual proof of how/why it works.
     
  2. WBahn

    Moderator

    Mar 31, 2012
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    What do you mean by "proof of Cramer's Rule for evaluating determinents or matrixes"?

    What does it mean to "evaluate a matrix"?

    How are you using Cramer's Rule to evaluate a determinant?
     
  3. Glenn Holland

    Thread Starter Member

    Dec 26, 2014
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    A rephrasing of my question is simply how/why does this rule provide a method of solving simultaneous equations.
     
  4. WBahn

    Moderator

    Mar 31, 2012
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  5. tjohnson

    Active Member

    Dec 23, 2014
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    For equations with only two unknown variables, the proof of Cramer's rule is fairly simple:

    (ax + by = e) * d = (dax + dby = de)<br />
(cx + dy = f) * -b = (-bcx - bdy = -bf)<br />
dax - bcx = (ad - bc)x = de - bf<br />
x = \frac{de - bf}{ad - bc} = \frac{\begin{bmatrix}e & b\\f & d\end{bmatrix}}{\begin{bmatrix}a & b\\c & d\end{bmatrix}}<br />
(ax + by = e) * -c = (-cax - cby = -ce)<br />
(cx + dy = f) * a = (acx + ady = af)<br />
ady - cby = (ad - bc)y = af - ce<br />
y = \frac{af - ce}{ad - bc} = \frac{\begin{bmatrix}a & e\\c & f\end{bmatrix}}{\begin{bmatrix}a & b\\c & d\end{bmatrix}}

    I had to dig out my math textbook to refresh my memory on this!:p

    EDIT: Corrected wrong math
     
    Last edited: Aug 10, 2015
  6. chukwuma

    New Member

    Feb 5, 2013
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    You can get a more general proof from Introduction to Calculus volume 2 by John Fritz and Richard Courant. Read through the second chapter. You'll find a proof of crammer's rule at the end of the chapter. The proof is satisfactorily rigorous, and applies to any n by n matrix
     
  7. Glenn Holland

    Thread Starter Member

    Dec 26, 2014
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    110
    Thanks.

    I hope "satisfactorily rigorous" doesn't mean I'm in for some root canal work, but I'll take a look.

    Is that book on line?
     
  8. Papabravo

    Expert

    Feb 24, 2006
    10,137
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    If you mean, can you buy it from Amazon, then yes.
    If you mean, can I download a PDF, then sadly, no it is not.
     
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