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Cramer's rule (Theorem)

Let $ Ax=b$ be the matrix form of a system of $ n$ linear equations in $ n$ unknowns, with $ x$ and $ b$ as $ n\times 1$ column vectors and $ A$ an $ n \times n$ matrix. If $ \det(A)\ne 0$, then this system has a unique solution, and for each $ i$ ( $ 1\le i \le n$) ,

$\displaystyle x_i = \frac{\det(M_i)}{\det(A)} $

where $ M_i$ is $ A$ with column $ i$ replaced by $ b$.



"Cramer's rule" is owned by akrowne.
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Attachments:
example of Cramer's rule (Example) by drini
proof of Cramer's rule (Proof) by rmilson
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Cross-references: column, solution, column vectors, linear equations, matrix
There are 5 references to this entry.

This is version 5 of Cramer's rule, born on 2001-10-29, modified 2002-09-21.
Object id is 626, canonical name is CramersRule.
Accessed 21930 times total.

Classification:
AMS MSC: 15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions)
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