Talk:Cramer's rule

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)

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)

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

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)

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

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)

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)

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)

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

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)

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)

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)

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)

Cramer's Rule is useless

I would dispute all of the (limited) claims for useful applications for Cramer's rule. The cost of applying the rule grows exponentially with dimension and I can't imagine why anyone would want to use it in practice.

Certainly, it has no role in the solution of linear systems, however small. The claim at the start of the article that "as no pivoting is needed, it is more efficient than Gaussian elimination for small matrices" is plainly wrong...pivoting is extremely efficient and improves the quality of the solutions. Would anyone like to claim a stability result for Cramer's rule? And the best way to compute the determinants used in Cramer's rule is to use Gaussian elimination, anyway. Furthermore, why would anyone use a SIMD machine to solve small linear systems? How on earth can you do this efficiently? The phrase "This formula is, however, of limited practical value for larger matrices" would be better stated as This formula is, however, of NO practical value" wherever it appears.

"Cramer's rule is of theoretical importance in that it gives an explicit expression for the solution of the system". Why is this important? Why is this better than x = A^{-1}b? Cramer's rule can be used to give an explicit representation of an individual component independently of all other components...I have heard this used as justification for its importance but, again, this could be done much more efficiently with Gaussian elimination (exploiting the resulting LU factorization).

Can anyone substantiate the line "Cramer's rule is also extremely useful for solving problems in differential geometry"? I would expect that anything it can do can be done better another way. But I am willing to be proved wrong.

Applications to algebra/relevance to eigenvalues: is it really worth mentioning that the rule can be used to prove well known results in an ugly way?

I apologize to the Cramer family for being so negative, but I have seen no evidence that the rule is anything more than a historical curiosity.

If no-one wants to refute my claims, I will happily edit the page to conform with my world view. Not that it will stop climate change, or anything :-) —Preceding unsigned comment added by 130.159.17.136 (talk) 17:18, 10 December 2009 (UTC)

First, wikipedia articles should not be written to confirm with any particular world view, but should reflect general consensus. If everyone considered Cramer's rule useless, then mankind would have forgotten about it long ago and this article would not have existed.

Now some more specific points. Although the greatest importance of Cramer's rule is definitely not in computer programming (and Cramer was not into computer science), it seems likely that, at least when one comes across a situation where a program needs the solution of a specific 2×2 system, the easiest solution will be to code the solution by Cramer's rule directly rather than to invoke some general equation-solving procedure involving Gaussian elimination. You say writing x = A^{-1}b is a theoretical alternative to Cramer's rule, but it is of no use unless you have an expression for A^{-1}. Such an expression exists... and is a blown-up version of Cramer's rule! In fact if you imagine any system where the coefficients are not completely known numbers (say they depend on one or more unspecified parameters; the worst case is that all coefficients are independent parameters), then one cannot apply Gaussian elimination or any other method that needs to make decisions depending on the values of the coefficients. Cramer's rule on the other hand needs to make no decisions at all (all that matters is that the denominator is nonzero, which is the condition for a unique solution to exist in the first place) and can be applied to such a system. That is exactly why the rule is of theoretical importance. There are many occasions in algebra where it is important for reasoning to know that an expression of a certain kind exists, even if one has no desire to actually use such an expression in computation. Marc van Leeuwen (talk) 08:23, 11 December 2009 (UTC)