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When did Gabriel Cramer come up with Cramer's Rule?

ANS: Around 1750, two years before his death. He died without giving an explanation of how exactly he came up with the rule. There is some speculation that Colin Maclaurin came up with the same rule in 1729 but there must have been a paper shortage in Scotland because it was never published until after he died.

A search on google for "cramer's rule maclaurin" comes up with several pages discussing who came up with it first. This one (http://www.findarticles.com/p/articles/mi_qa3789/is_200110/ai_n8969722) says that Maclaurin published it in 1748, two years before Cramer did, but that Cramer's probably gained popularity due to his superior notation.

More precise statement

Any chance someone is willing to give a more precise statement of Cramer's rule? As in the following: Let R be a commutative ring, M a finitely generated R-module with generators $m_{i}$ , and $\phi :M\rightarrow M$ an endomorphism of R-modules. $\phi$ determines a matrix

$A=(a_{ij})$

via $\phi (m_{i})=\sum a_{ij}m_{j}$ . Then Adj(A)A=det(A)I, where Adj(A) is the cofactor matrix of A.

That is not Cramer's rule, though it is related, but in a not so trivial way. There is no general rule for solving square linear systems over a commutative ring for the simple reason that they generally do not have unique solutions (try solving

3x=b

in Z/6Z). Something can be said (if the determinant of the system happens to be an invertible element of the ring for instance) but I'm not sure this article is the right place. Marc van Leeuwen (talk) 14:55, 7 September 2009 (UTC)[reply]

Does the example of comparative statics in econ make any sense?

am I right that there's nothing special about comparative statics, it's just an example of solving a system of linear equations, nothing particularly about cramer's rule. This latest edit says it's of 'further significance', but it doesn't seem to be. --Tristanreid

I don't find that example much useful either. Oleg Alexandrov (talk) 23:52, 17 October 2005 (UTC)[reply]

ok, I'm yanking it. Thanks for signing me, by the way. I forgot last time. Tristanreid 02:44, 19 October 2005 (UTC)[reply]

Where did Gaon come from?

Vilna Gaon seems to have come out of left field. I note that the main article on him includes a similar statement without attribution. Does anyone have a reference for this statement? S.N. Hillbrand 20:45, 22 December 2005 (UTC)[reply]

Thanks for removing that. Looked like a strange edit. Oleg Alexandrov (talk) 01:25, 23 December 2005 (UTC)[reply]

I actually remember it being printed in my algebra book as a possible etymology. I really don't know if there is a credible source for it. I think they might have cast doubt on it themselves. I have no idea how it even got started, but unless I am mistaken, it was in my HBJ algebra book issued by the Los Angeles Unified School District. PhatJew 19:18, 9 March 2006 (UTC)[reply]

The reference to the Gaon is a legitimate one that is quoted in many sources. It should be kept as an alternative entymology.

Where has the nice proof gone?

About a year ago there was a beautiful proof of the Cramer's rule on this wikipedia page. Can you, please, bring it back? Thanks Pavel.Pokorny@vscht.cz —Preceding unsigned comment added by 147.33.113.54 (talk) 17:05, 6 December 2007 (UTC)[reply]

Incorrect Example

The 4th image in the example section is incorrect. The determinant in the numerator is calculated in correctly. Sorry I cannot fix it right now,will someone else?--128.101.152.71 (talk) 21:12, 10 March 2008 (UTC)[reply]

Done. Michael Slone (talk) 01:32, 11 March 2008 (UTC)[reply]

Re: Relevance to eigenvalues

"The numerator will always be zero, since here c=0."

I can't find any previous reference to c. Is this a remnant of an earlier edit? -- 72.224.136.152 (talk) 14:41, 23 September 2009 (UTC)[reply]

OK, the problem came from an earlier edit. Variable c was changed to b but a later reference to it remained unchanged. -- TheMaestro (talk) 02:08, 24 September 2009 (UTC)[reply]

All right, thank you. But here is another question: what does this section have to do in this article in the first place? Cramer's rule is about the case where the coefficient determinant is non-zero, in which case it gives an explicit expression for the solution. But in the case of homogeneous linear equations this is quite unnecessary: it is obvious that the zero vector will always be a solution, so you don't need a complicated formula to tell you that in case the coefficient determinant is non-zero. The only relevant point here is that the solution will be non-unique precisely when that determinant is zero; although that could possibly be considered to be a (minor) part of the statement of the theorem called Cramer's rule, it is not even mentioned in the current article. (If it were, it should be mentioned more precisely that coefficient determinant zero implies either a non-unique solution for the whole system or no solution at all, since the article talks about non-homogeneous systems; also beware that when the determinant is zero there could still be a unique solution for some of the unknowns.) My feeling is that this subsection does not belong here, and should be removed, or possibly moved to an article is it more relevant to. Marc van Leeuwen (talk) 06:50, 24 September 2009 (UTC)[reply]

I agree, I think the connection to Cramer's rule is tangential at best, in more senses than one. -- TheMaestro (talk) 01:50, 25 September 2009 (UTC)[reply]

Revision as of 06:50, 24 September 2009 edit Marc van Leeuwen (talk \| contribs) Extended confirmed users, Pending changes reviewers 2,666 edits →Re: Relevance to eigenvalues: Raised a more fundamental question ← Previous edit		Revision as of 01:50, 25 September 2009 edit undo TheMaestro (talk \| contribs) 330 edits No edit summary Next edit →
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	All right, thank you. But here is another question: what does this section have to do in this article in the first place? Cramer's rule is about the case where the coefficient determinant is non-zero, in which case it gives an explicit expression for the solution. But in the case of ''homogeneous'' linear equations this is quite unnecessary: it is obvious that the zero vector will always be a solution, so you don't need a complicated formula to tell you that in case the coefficient determinant is non-zero. The only relevant point here is that the solution will be non-unique ''precisely'' when that determinant is zero; although that could possibly be considered to be a (minor) part of the statement of the theorem called Cramer's rule, it is not even mentioned in the current article. (If it were, it should be mentioned more precisely that coefficient determinant zero implies either a non-unique solution for the whole system or no solution at all, since the article talks about non-homogeneous systems; also beware that when the determinant is zero there could still be a unique solution for ''some'' of the unknowns.) My feeling is that this subsection does not belong here, and should be removed, or possibly moved to an article is it more relevant to. [[User:Marc van Leeuwen\|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen\|talk]]) 06:50, 24 September 2009 (UTC)		All right, thank you. But here is another question: what does this section have to do in this article in the first place? Cramer's rule is about the case where the coefficient determinant is non-zero, in which case it gives an explicit expression for the solution. But in the case of ''homogeneous'' linear equations this is quite unnecessary: it is obvious that the zero vector will always be a solution, so you don't need a complicated formula to tell you that in case the coefficient determinant is non-zero. The only relevant point here is that the solution will be non-unique ''precisely'' when that determinant is zero; although that could possibly be considered to be a (minor) part of the statement of the theorem called Cramer's rule, it is not even mentioned in the current article. (If it were, it should be mentioned more precisely that coefficient determinant zero implies either a non-unique solution for the whole system or no solution at all, since the article talks about non-homogeneous systems; also beware that when the determinant is zero there could still be a unique solution for ''some'' of the unknowns.) My feeling is that this subsection does not belong here, and should be removed, or possibly moved to an article is it more relevant to. [[User:Marc van Leeuwen\|Marc van Leeuwen]] ([[User talk:Marc van Leeuwen\|talk]]) 06:50, 24 September 2009 (UTC)
			:I agree, I think the connection to Cramer's rule is tangential at best, in more senses than one. -- [[User:TheMaestro\|TheMaestro]] ([[User talk:TheMaestro\|talk]]) 01:50, 25 September 2009 (UTC)