example of Cramer’s rule


Say we want to solve the system

3x+2y+z-2w = 4
2x-y+2z-5w = 15
4x+2y-5w = 1
3x-2z-4w = 1.

The associated matrix is

(321-22-12-5420-130-2-4)

whose determinantMathworldPlanetmath is Δ=-65. Since the determinant is non-zero, we can use Cramer’s rule. To obtain the value of the k-th variable, we replace the k-th column of the matrix above by the column vectorMathworldPlanetmath

(41511),

the determinant of the obtained matrix is divided by Δ and the resulting value is the wanted solution.

So

x=Δ1Δ=|421-215-12-5120-110-2-4|-65=-65-65=1
y=Δ2Δ=|341-22152-5410-131-2-4|-65=130-65=2
z=Δ3Δ=|324-22-115-54211301-4|-65=-195-65=3
w=Δ4Δ=|32142-1215420130-21|-65=65-65=-1
Title example of Cramer’s rule
Canonical name ExampleOfCramersRule
Date of creation 2013-03-22 12:50:33
Last modified on 2013-03-22 12:50:33
Owner drini (3)
Last modified by drini (3)
Numerical id 7
Author drini (3)
Entry type Example
Classification msc 15A15