The following sections of this BookRags Literature Study Guide is offprint from Gale's For Students Series: Presenting Analysis, Context, and Criticism on Commonly Studied Works: Introduction, Author Biography, Plot Summary, Characters, Themes, Style, Historical Context, Critical Overview, Criticism and Critical Essays, Media Adaptations, Topics for Further Study, Compare & Contrast, What Do I Read Next?, For Further Study, and Sources.
(c)1998-2002; (c)2002 by Gale. Gale is an imprint of The Gale Group, Inc., a division of Thomson Learning, Inc. Gale and Design and Thomson Learning are trademarks used herein under license.
The following sections, if they exist, are offprint from Beacham's Encyclopedia of Popular Fiction: "Social Concerns", "Thematic Overview", "Techniques", "Literary Precedents", "Key Questions", "Related Titles", "Adaptations", "Related Web Sites". (c)1994-2005, by Walton Beacham.
The following sections, if they exist, are offprint from Beacham's Guide to Literature for Young Adults: "About the Author", "Overview", "Setting", "Literary Qualities", "Social Sensitivity", "Topics for Discussion", "Ideas for Reports and Papers". (c)1994-2005, by Walton Beacham.
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Cramer's rule gives a formula for the entries of a column vector x that satisfies Ax = b, where A is a n x n matrix with nonzero determinant and b is a column vector with n entries. A column vector in n x 1 matrix. Let Aj(b) denote the n x n matrix which has the same entries as A except for the jth column which is equal to b. Cramer's rule is that the jth entry of x must be equal to the det(Aj(b))/det A. The notation det(M) means the determinant of the matrix M.
Here is the proof of Cramer's rule. Let ej be the length n column vector that has jth entry equal to one and all other entries equal to zero. The identity matrix is [e1 e2 ... en] = I. AIj (x) = [Ae1 ... Ax ... Aen] = Aj (b). Because the determinant of a product of matrices is equal to the product of their determinants, det(A)*det(Ij(x))=det(Aj(b)). Using cofactor expansion, it is easy to see that det(Ij(x)) = the jth entry of x. Therefore the jth entry of x = det(Aj(b))/det(A).
Cramer's rule is not very practical as an algorithm for solving the equation Ax = b because too many computations are involved. Generally, however, there is some margin of error in figuring b. So, it is often useful to know all the solutions for Ax = b as b varies over its margin of error. In this case, Cramer's rule is helpful for understanding the set of solutions.