Cramer's Rule

A precise determinant formula for square systems, with invertibility conditions and computational limits.

Represent → transform → verify. A linear-algebra object is represented as coordinates or a matrix, transformed through valid row or column operations, and checked against the original invariant or equation.

Precise definition

For a square system $A\mathbf x=\mathbf b$ with $\det A\ne0$, Cramer's rule gives $x_i=\det A_i/\det A$, where $A_i$ is obtained by replacing column $i$ of $A$ by $\mathbf b$. The nonzero determinant is essential: it means the system has a unique solution.

Notation and mathematical language

Column replacement matches the position of the unknown because $A\mathbf x$ is a linear combination of the columns of $A$. Cramer's rule applies to $n$ equations in $n$ unknowns. It is a theoretical identity and can be convenient for tiny symbolic systems.

Conceptual picture

Multilinearity and alternation of the determinant isolate one coefficient when a column is replaced by the right-hand side. Every other term repeats a column and has determinant zero; the surviving term is $x_i\det A$.

Fully worked example

Interpretation and application

Cramer's rule shows each solution component depends rationally on entries of $A$ and $\mathbf b$. This is useful in proofs and sensitivity formulas, but near-zero determinants can magnify perturbations; determinant size alone is not a scale-invariant condition number.