GraphMath

Cramer's rule

Determinant ratios, signed volumes and rank-deficient cases

Why does replacing one column of A reveal one solution coordinate?

Cramer's rule expresses each component of the solution to Ax = b as a determinant ratio. This chapter derives the formula algebraically, explains the signed-volume geometry in ℝ³ and ℝⁿ, then looks at what the same determinant-ratio picture says for rank-deficient inconsistent and consistent systems.

Open chapter PDF Key ideas Chapter index Back to Linear Algebra

Key ideas

Cramer's rule turns coordinates into ratios of determinants.

  • For a square full-rank matrix A, each solution component xk is given by |Ak| / |A|
  • The numerator matrix Ak is obtained from A by replacing column k with b
  • The algebraic derivation builds a matrix Ck with det(Ck) = xk and uses |A Ck| = |A| |Ck|
  • In ℝ³, the ratio can be read as a ratio of signed heights over the same base, or equivalently signed volumes
  • In ℝⁿ, the same idea uses the hyperplane spanned by all columns except ai and a normal direction to that hyperplane
  • If |A| = 0, the usual Cramer ratio is not defined, but the determinant/area picture still explains inconsistent and consistent rank-deficient behavior

The visual sections show how determinant ratios behave in full-rank 3D systems and what changes when the denominator volume collapses to zero.

Why does replacing one column isolate one coordinate?

In the full-rank case, b can be written uniquely as x₁a₁ + ⋯ + xₙaₙ. When column k is replaced by b, determinant multilinearity keeps only the term containing xkak; all other terms have repeated or dependent columns. Thus |Ak| = xk|A|.

Related chapters

  • Determinant develops the determinant rules used here, especially the product rule and the geometric interpretation of signed area and volume
  • Determinant visualization supports the signed-volume and orientation interpretation behind the signs of the Cramer ratios
  • Linear equations provides the system Ax = b viewpoint and the structural meaning of solvability
  • Inverse covers the square full-rank case in which Cramer's rule gives the unique solution
  • Change of basis explains the coordinate conversion used here to reduce rank-2 area-ratio calculations to a simpler 2D coordinate system

Examples with visualizations

Cramer's rule in 3D

Animated determinant ratios and sign interpretation for x₁, x₂ and x₃

Animated 3D illustration of Cramer's rule showing determinant ratios for x₁, x₂ and x₃ and the sign of each coordinate
Open visual page Open in PDF

Why Cramer's rule works

Geometric derivation in ℝ³ and ℝⁿ using signed volumes and determinant ratios

Thumbnail for the geometric derivation of Cramer's rule in ℝ³ using signed volumes
Open visual page Open in PDF

Chapter contents

The PDF is a single document. The page links below are best-effort: most browsers support them, but some viewers may ignore the page hint.

Topic Pages
Algebraic derivation 1–3
Geometric derivation in ℝ³ 3–7
Geometric derivation in ℝⁿ 7–9
3D illustration for a full-rank system 9–11
Inconsistent Ax = b: rank-2 A, b ∉ col(A) 11–13
Consistent Ax = b: rank-2 A, b ∈ col(A) 13–17
Open PDF

What changes when A is rank-deficient?

Then |A| = 0, so the usual Cramer ratio is not defined. If b is outside col(A), the denominator volume is zero while some numerator volumes are nonzero, indicating inconsistency. If b lies in col(A), the numerator and denominator triples are all flat; the solution family can then be described by area ratios inside the common plane.

Was this chapter helpful?

Quick feedback helps us improve the site.

Yes No