Cramer's rule - Wikipedia

Abstraction: Explicit determinant-based formula solving systems of linear equations

Key points:

• Expresses each unknown xᵢ as det(Aᵢ)/det(A), where Aᵢ is A with the i-th column replaced by the right-hand-side vector b
• Named after Gabriel Cramer (1750); Colin Maclaurin published special cases in 1748 and possibly knew of it by 1729
• Valid only when det(A) ≠ 0 (unique solution exists); zero determinant means inconsistent or indeterminate system
• Computationally inefficient for n > 3 — requires n+1 determinants vs single-pass Gaussian elimination; mainly a theoretical tool
• Used in Ricci calculus for Christoffel symbol computations and proving divergence operator coordinate invariance
• Applied to derive variation of parameters solution for inhomogeneous linear differential equations

Connections: Linear Algebra · Determinants · Linear Systems

Source: http://en.wikipedia.org/wiki/Cramer%27s_rule