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.

 

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?

 

A rephrasing of my question is simply how/why does this rule provide a method of solving simultaneous equations.

 

A few different proofs are shown on Wikipedia:

https://en.wikipedia.org/wiki/Cramer's_rule#Other_proofs

 

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

 

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

 

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

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?

 

...
Is that book on line?

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.