Cramer's rule

Cramer's rule expresses each solution coordinate of a square linear system as a ratio of determinants, provided the coefficient matrix has nonzero determinant.

Determinant: A scalar associated with a square matrix that encodes properties including invertibility and signed volume scaling. The denominator and numerators in Cramer's rule are determinants.
System of linear equations: A collection of linear equations required to hold for the same unknown variables. Cramer's rule solves this kind of system when the number of equations matches the number of unknowns.
Two-by-two linear system: A pair of linear equations in two unknowns. Cramer's rule reduces this common case to three small determinants.
Gaussian elimination: A sequence of row operations that reduces a linear system to an equivalent, simpler form. It solves systems efficiently without computing a separate determinant for every variable.
Gabriel Cramer: A Swiss mathematician whose 1750 work presented the determinant rule now bearing his name. The rule is named after Cramer's published formulation.
Cofactor expansion: A method for computing a determinant by expanding along a row or column using signed minors. It can evaluate the determinants used in each coordinate formula.
Coefficient matrix: The matrix whose entries are the coefficients of the variables in a linear system. Its determinant decides whether the rule's denominator is nonzero.
Three-by-three linear system: A system of three linear equations in three unknowns. It illustrates the coordinate formulas before determinant expansion becomes unwieldy.
LU decomposition: A factorization of a matrix into lower- and upper-triangular matrices, often used to solve linear systems. It reuses matrix factors across multiple right-hand sides, unlike repeated Cramer determinants.
Colin Maclaurin: A Scottish mathematician who published determinant-based solutions of linear systems before Cramer's work. His earlier treatment shows that the method predates its familiar name.

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