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
469
161
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
20,059
5,655
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
469
161
A rephrasing of my question is simply how/why does this rule provide a method of solving simultaneous equations.

Mar 31, 2012
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5. tjohnson Active Member

Dec 23, 2014
625
122
For equations with only two unknown variables, the proof of Cramer's rule is fairly simple:

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

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

EDIT: Corrected wrong math

Last edited: Aug 10, 2015
6. chukwuma New Member

Feb 5, 2013
1
0
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
469
161
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
11,076
2,152
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.