Cramer's Rule — Determinant Ratios and 3D Geometry

A visual note on solving a square full-rank system by replacing one column at a time

Cramer's rule expresses each component of the solution to A x = b as a ratio of determinants. For a square full-rank matrix A, the solution is unique, and each coordinate xk can be recovered by replacing the k-th column of A with b.

Geometrically, the determinant measures signed volume. The numerator determinant records the signed volume after one column has been replaced by the target vector b, and the denominator determinant records the signed volume of the original column parallelepiped.

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Cramer's rule in 3D

Animated 3D illustration of Cramer's rule showing determinant ratios for x1, x2 and x3 and the sign of each coordinate.
The animation shows how each coordinate x₁, x₂ and x₃ is computed from a determinant ratio. Replacing one column of A by b produces the numerator matrix, and the sign of the resulting determinant is interpreted geometrically through signed volume.

Concept

In a system A x = b, the columns of A form the directions used to build b as a linear combination. Cramer's rule isolates one coordinate at a time by replacing the corresponding column of A with b.

If Ak is the matrix obtained from A by replacing column k with b, then the ratio |Ak| / |A| gives the coordinate xk. The determinant product rule is what makes this isolation possible.

Structure

The denominator |A| measures the signed volume of the original parallelepiped formed by the columns of A. The numerator |Ak| measures the signed volume after one generating vector has been replaced by b.

Dividing these signed volumes extracts one solution coordinate. In 3D, the sign of that coordinate can be read from which side of the opposite face the replaced vector lies on, together with the orientation of the original column frame.

Key equations

A x = b
|A| ≠ 0
Ak = A with column k replaced by b
xk = |Ak| / |A|
x₁ = |A₁| / |A|
x₂ = |A₂| / |A|
x₃ = |A₃| / |A|

These equations express each solution coordinate as a ratio of signed volumes.

Query phrases

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Reference

Related chapter: Cramer's rule