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

Gauss Jordan Matrix Equations Elimination Method Step by Step Example

Jordan Matrix Elimination Method Step by Step Example

Jordan 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

Find the pivot in the 1st column and swap the 2nd and the 1st rows

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

Multiply the 1st row by 3

	x	y	z	b
1	3	-3	6	12
2	3	2	-1	3
3	2	3	-1	3

Subtract the 1st row from the 2nd row and restore it

	x	y	z	b
1	1	-1	2	4
2	0	5	-7	-9
3	2	3	-1	3

Multiply the 1st row by 2

	x	y	z	b
1	2	-2	4	8
2	0	5	-7	-9
3	2	3	-1	3

Subtract the 1st row from the 3rd row and restore it

	x	y	z	b
1	1	-1	2	4
2	0	5	-7	-9
3	0	5	-5	-5

Make the pivot in the 2nd column by dividing the 2nd row by 5

	x	y	z	b
1	1	-1	2	4
2	0	1	-1.4	-1.8
3	0	5	-5	-5

Multiply the 2nd row by -1

	x	y	z	b
1	1	-1	2	4
2	0	-1	1.4	1.8
3	0	5	-5	-5

Subtract the 2nd row from the 1st row and restore it

	x	y	z	b
1	1	0	0.6000000000000001	2.2
2	0	1	-1.4	-1.8
3	0	5	-5	-5

Multiply the 2nd row by 5

	x	y	z	b
1	1	0	0.6000000000000001	2.2
2	0	5	-7	-9
3	0	5	-5	-5

Subtract the 2nd row from the 3rd row and restore it

	x	y	z	b
1	1	0	0.6000000000000001	2.2
2	0	1	-1.4	-1.8
3	0	0	2	4

Make the pivot in the 3rd column by dividing the 3rd row by 2

	x	y	z	b
1	1	0	0.6000000000000001	2.2
2	0	1	-1.4	-1.8
3	0	0	1	2

Multiply the 3rd row by 0.6000000000000001

	x	y	z	b
1	1	0	0.6000000000000001	2.2
2	0	1	-1.4	-1.8
3	0	0	0.6000000000000001	1.2000000000000002

Subtract the 3rd row from the 1st row and restore it

	x	y	z	b
1	1	0	0	1
2	0	1	-1.4	-1.8
3	0	0	1	2

Multiply the 3rd row by -1.4

	x	y	z	b
1	1	0	0	1
2	0	1	-1.4	-1.8
3	0	0	-1.4	-2.8

Subtract the 3rd row from the 2nd row and restore it

	x	y	z	b
1	1	0	0	1
2	0	1	0	0.9999999999999998
3	0	0	1	2

Solution set:

x1 = 1

x2 = 0.9999999999999998

x3 = 2