page_title: Cramer's rule
section_title: Geometric interpretation
hierarchical_section_title: Cramer's rule / Geometric interpretation
caption_attribution_description: Geometric interpretation of Cramer's rule. The areas of the second and third shaded parallelograms are the same and the second is x_1 times the first. from this equality Cramer's rule follows.
context_page_description: In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand-sides of the equations. It is named after Gabriel Cramer, who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748.
  Cramer's rule implemented in a naïve way is computationally inefficient for systems of more than two or three equations. In the case of n equations in n unknowns, it requires computation of n + 1 determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single determinant. Cramer's rule can also be numerically unstable even for 2×2 systems.
context_section_description: Cramer's rule has a geometric interpretation that can be considered also a proof or simply giving insight about its geometric nature. These geometric arguments work in general and not only in the case of two equations with two unknowns presented here.
  Given the system of equations
  it can be considered as an equation between vectors
  The area of the parallelogram determined by  and  is given by the determinant of the system of equations:
  In general, when there are more variables and equations, the determinant of n vectors of length n will give the volume of the parallelepiped determined by those vectors in the n-th dimensional Euclidean space.
  Therefore, the area of the parallelogram determined by  and  has to be  times the area of the first one since one of the sides has been multiplied by this factor. Now, this last parallelogram, by Cavalieri's principle, has the same area as the parallelogram determined by  and 
  Equating the areas of this last and the second parallelogram gives the equation
  from which Cramer's rule follows.
caption: Geometric interpretation of Cramer's rule. The areas of the second and third shaded parallelograms are the same and the second is 
    
      
        
          
            x
            
              1
            
          
        
      
      {\displaystyle x_{1}}
    
   times the first. From this equality Cramer's rule follows.
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