Cramer's rule

Cramer's rule is a theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants.

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.

It is named after Gabriel Cramer (1704 - 1752).

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

$Ax=c$

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:

$x_{i}={\det(A_{i}) \over \det(A)}$

where $A_{i}$ is the matrix formed by replacing the ith column of $A$ by the column vector $c$.

Example

A good way to use Cramer's Rule on a 2x2 matrix is to use this formula:

Given

$ax+by=e$ and

$cx+dy=f$,

$x={ed-bf \over ad-bc}$

$y={af-ec \over ad-bc}$