Cramer's rule is a theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants. It is named after Gabriel Cramer (1704 - 1752). A minority attribute it to Vilna Gaon Eliyahu Cramer (April 23, 1720 – October 9, 1797).

Computationally, it is generally inefficient and thus not used in practical applications which may involve many equations. However, it is of theoretical importance in that it gives an explicit expression for the solution of the system.

Elementary formulation

---

The system of equations is represented in matrix multiplication form as:

$Ax\; =\; c\backslash ,$

where the square matrix $A$ is invertible and the vector $x$ is the column vector of the variables: $(\; x\_i\; )$.

The theorem then states that:

$x\_i\; =\; \{\; \backslash det(A\_i)\; \backslash over\; \backslash det(A)\}$

where $A\_i$ is the matrix formed by replacing the ith column of $A$ by the column vector $c$.

Abstract formulation

---

Let R be a commutative ring, A an n×n matrix with coefficients in R. Then

$\backslash mathrm\{Adj\}(A)A\; =\; \backslash mathrm\{det\}(A)I\backslash ,$

where Adj(A) denotes the adjugate of A, det(A) is the determinant, and I is the identity matrix.

Example

---

A good way to use Cramer's rule on a 2×2 matrix is to use this formula:

Given

$ax\; +\; by\; =\; e\backslash ,$ and

$cx\; +\; dy\; =\; f\backslash ,$,

which in matrix format is

x and y can be found with Cramer's rule as:

and

The rules for 3×3 are similar. Given

$ax\; +\; by\; +\; cz\; =\; j\backslash ,$,

$dx\; +\; ey\; +\; fz\; =\; k\backslash ,$ and

$gx\; +\; hy\; +\; iz\; =\; l\backslash ,$,

which in matrix format is

x, y and z can be found like so:

, , and

Applications to differential geometry

---

Cramer's rule is also extremely useful for solving problems in differential geometry. Consider the two equations $F(x,\; y,\; u,\; v)\; =\; 0\backslash ,$ and $G(x,\; y,\; u,\; v)\; =\; 0\backslash ,$. When u and v are independent variables, we can define $x\; =\; X(u,\; v)\backslash ,$ and $y\; =\; Y(u,\; v)\backslash ,$.

Finding an equation for $\backslash partial\; x/\backslash partial\; u$ is a trivial application of Cramer's rule.

First, calculate the first derivatives of F, G, x and y.

$dF\; =\; \backslash frac\{\backslash partial\; F\}\{\backslash partial\; x\}\; dx\; +\; \backslash frac\{\backslash partial\; F\}\{\backslash partial\; y\}\; dy\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; u\}\; du\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; v\}\; dv\; =\; 0$

$dG\; =\; \backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\}\; dx\; +\; \backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\; dy\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; u\}\; du\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; v\}\; dv\; =\; 0$

$dx\; =\; \backslash frac\{\backslash partial\; X\}\{\backslash partial\; u\}\; du\; +\; \backslash frac\{\backslash partial\; X\}\{\backslash partial\; v\}\; dv$

$dy\; =\; \backslash frac\{\backslash partial\; Y\}\{\backslash partial\; u\}\; du\; +\; \backslash frac\{\backslash partial\; Y\}\{\backslash partial\; v\}\; dv$

Substituting dx, dy into dF and dG, we have:

$dF\; =\; \backslash left(\backslash frac\{\backslash partial\; F\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; u\}\; \backslash right)\; du\; +\; \backslash left(\backslash frac\{\backslash partial\; F\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; v\}\; \backslash right)\; dv\; =\; 0$

$dG\; =\; \backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; u\}\; \backslash right)\; du\; +\; \backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; v\}\; \backslash right)\; dv\; =\; 0$

Since u, v are both independent, the coefficients of du, dv must be zero. So we can write out equations for the coefficients:

$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; u\}\; =\; -\backslash frac\{\backslash partial\; F\}\{\backslash partial\; u\}$

$\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; u\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; u\}\; =\; -\backslash frac\{\backslash partial\; G\}\{\backslash partial\; u\}$

$\backslash frac\{\backslash partial\; F\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; F\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; v\}\; =\; -\backslash frac\{\backslash partial\; F\}\{\backslash partial\; v\}$

$\backslash frac\{\backslash partial\; G\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; x\}\{\backslash partial\; v\}\; +\backslash frac\{\backslash partial\; G\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; y\}\{\backslash partial\; v\}\; =\; -\backslash frac\{\backslash partial\; G\}\{\backslash partial\; v\}$

Now, by Cramer's rule, we see that:

$$

\frac{\partial x}{\partial u} = \frac{\begin{vmatrix} -\frac{\partial F}{\partial u} & \frac{\partial F}{\partial y} \\ -\frac{\partial G}{\partial u} & \frac{\partial G}{\partial y}\end{vmatrix}}{\begin{vmatrix}\frac{\partial F}{\partial x} & \frac{\partial F}{\partial y} \\ \frac{\partial G}{\partial x} & \frac{\partial G}{\partial y}\end{vmatrix}}

This is now a formula in terms of two Jacobians:

$\backslash frac\{\backslash partial\; x\}\{\backslash partial\; u\}\; =\; -\; \backslash frac\{\backslash left(\backslash frac\{\backslash partial\backslash left(F,\; G\backslash right)\}\{\backslash partial\backslash left(y,\; u\backslash right)\}\backslash right)\}\{\backslash left(\backslash frac\{\backslash partial\backslash left(F,\; G\backslash right)\}\{\backslash partial\backslash left(x,\; y\backslash right)\}\backslash right)\}$

Similar formulae can be derived for $\backslash frac\{\backslash partial\; x\}\{\backslash partial\; v\}$, $\backslash frac\{\backslash partial\; y\}\{\backslash partial\; u\}$, $\backslash frac\{\backslash partial\; y\}\{\backslash partial\; v\}$.

Applications to algebra

---

Cramer's rule can be used to prove the Cayley-Hamilton theorem of linear algebra, as well as Nakayama's lemma, which is fundamental in commutative ring theory.

Linear algebra | Mathematical theorems

Cramersche Regel | Regla de Cramer | Règle de Cramer | 크래머 공식 | Regola di Cramer | Wzory Cramera | Cramers regel