Cramer's Rule to Solve a System of 3 Linear Equation with Step by Step Example

How to Solve a System of 3x3 Matrix Equations using Cramer's Rule with Step by Step Example

Cramer's Rule Matrix method can be used to solve systems of linear equations involving two or more variables.

Example:
3x + 2y - z = 3
x - y + 2z = 4
2x + 3y - z = 3

Solution:

Given matrix

	x	y	z	b
1	3	2	-1	3
2	1	-1	2	4
3	2	3	-1	3

Write down the main matrix and find its determinant,

	x	y	z
1	3	2	-1
2	1	-1	2
3	2	3	-1

Determinant D = -10

Replace the 1st column of the main matrix with the solution vector and find its determinant

	x	y	z
1	3	2	-1
2	4	-1	2
3	3	3	-1

D1 = -10

Replace the 2nd column of the main matrix with the solution vector and find its determinant

	x	y	z
1	3	3	-1
2	1	4	2
3	2	3	-1

D2 = -10

Replace the 3rd column of the main matrix with the solution vector and find its determinant

	x	y	z
1	3	2	3
2	1	-1	4
3	2	3	3

D3 = -20

x1 = D1 / D = -10 / (-10) = 1

x2 = D2 / D = -10 / (-10) = 1

x3 = D3 / D = -20 / (-10) = 2

Solution:

x = 1

y = 1

z = 2