Cramer's Rule

Statement

Let $A$ be an invertible $n \times n$ matrix over a commutative ring $R$ (so $\det(A)$ is a unit), and let $b \in R^n$. The unique solution $x$ to $Ax = b$ satisfies:

where $A_i(b)$ is the matrix $A$ with its $i$-th column replaced by $b$.

In the non-invertible / commutative ring setting, the adjugate form of the rule states:

where $\operatorname{cramer}(A, b)_i = \det(A_i(b))$, without requiring $\det(A)$ to be a unit.

Visualization

Solve the $2 \times 2$ system:

which corresponds to $Ax = b$ with:

Compute $\det(A)$:

Since $\det(A) = 5 \neq 0$, the system has a unique solution.

Compute $\det(A_1(b))$ — replace column 1 with $b$:

Compute $\det(A_2(b))$ — replace column 2 with $b$:

Apply Cramer's rule:

Verify:

Proof Sketch

The adjugate satisfies $A \cdot \operatorname{adj}(A) = \det(A) \cdot I$ by definition (cofactor expansion). Setting $\operatorname{cramer}(A, b) = \operatorname{adj}(A) \cdot b$, one computes:

The $i$-th component of $\operatorname{adj}(A) \cdot b$ is $\sum_j (-1)^{i+j} \det(A_{ji})\, b_j$, which is precisely the cofactor expansion of $\det(A_i(b))$ along column $i$. When $\det(A)$ is a unit, dividing through gives the classical formula $x_i = \det(A_i(b))/\det(A)$.

Connections

• Rank–Nullity TheoremRank–Nullity Theorem$$\operatorname{rank}(f) + \operatorname{nullity}(f) = \dim V$$The dimensions of image and kernel of a linear map sum to the domain dimensionRead more → — Cramer's rule applies when $\operatorname{nullity}(A) = 0$; otherwise the system has no unique solution.
• Determinant MultiplicativityDeterminant Multiplicativity$$\det(AB) = \det(A)\,\det(B)$$The determinant of a product equals the product of determinantsRead more → — the adjugate identity $A \cdot \operatorname{adj}(A) = \det(A) I$ is a consequence of cofactor expansion and the multiplicativity of $\det$.
• Cayley–Hamilton TheoremCayley–Hamilton Theorem$$p_A(A) = 0 \;\text{where}\; p_A(\lambda) = \det(\lambda I - A)$$Every square matrix satisfies its own characteristic polynomialRead more → — the adjugate $\operatorname{adj}(A)$ appears in the Cayley–Hamilton proof; $p_A(A) = 0$ is assembled from the identity $(\lambda I - A)\operatorname{adj}(\lambda I - A) = p_A(\lambda)I$.
• Spectral TheoremSpectral Theorem$$A = Q \Lambda Q^T \;\text{(real symmetric case)}$$Every real symmetric matrix is orthogonally diagonalizableRead more → — in an orthogonal basis of eigenvectors, the linear system decouples and each component's solution is $b_i/\lambda_i$, a spectral analogue of Cramer's formula.
• Quadratic FormulaQuadratic Formula$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$Solve any quadratic equation using the discriminantRead more → — for $n=1$, Cramer's rule recovers $x = b/a$; the quadratic formula solves the $n=2$ eigenvalue equation.

/-- Cramer's rule in adjugate form: A *ᵥ (cramer A b) = det(A) • b.
    This holds over any commutative ring without requiring det(A) to be
    a unit. Mathlib locates this in LinearAlgebra.Matrix.Adjugate. -/
theorem cramers_rule {n R : Type*}
    [Fintype n] [DecidableEq n] [CommRing R]
    (A : Matrix n n R) (b : n → R) :
    A *ᵥ (Matrix.cramer A b) = A.det • b :=
  Matrix.mulVec_cramer A b