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

Example

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Given

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:ax+by = e and

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:cx+dy = f,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

which in matrix format is

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:egin{vmatrix} a & b \ c & d end{vmatrix}egin{vmatrix} x \ y end{vmatrix}=egin{vmatrix} e \ f end{vmatrix}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:x = rac { egin{vmatrix} e & b \ f & d end{vmatrix} } { egin{vmatrix} a & b \ c & d end{vmatrix} } = { ed - bf over ad - bc}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:and

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:y = rac { egin{vmatrix} a & e \ c & f end{vmatrix} } { egin{vmatrix} a & b \ c & d end{vmatrix} } = { af - ec over ad - bc}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~