高斯消去法

高斯消去法

Ax=b

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E1 :   -x1   +   x2   +  2x3   = 2
E2 :   3x1   -   x2   +    x3   = 6

E3 :   -x1   +   3x2 +  4x3   = 4

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利用 online gaussian elimination calculator
https://matrix.reshish.com/gauss-jordanElimination.php

Your matrix

	X1	X2	X3	b
1	-1	1	2	2
2	3	-1	1	6
3	-1	3	4	4

Find the pivot in the 1st column in the 1st row (inversing the sign in the whole row)

	X1	X2	X3	b
1	1	-1	-2	-2
2	3	-1	1	6
3	-1	3	4	4

Multiply the 1st row by 3

	X1	X2	X3	b
1	3	-3	-6	-6
2	3	-1	1	6
3	-1	3	4	4

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

	X1	X2	X3	b
1	1	-1	-2	-2
2	0	2	7	12
3	-1	3	4	4

Multiply the 1st row by -1

	X1	X2	X3	b
1	-1	1	2	2
2	0	2	7	12
3	-1	3	4	4

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

	X1	X2	X3	b
1	1	-1	-2	-2
2	0	2	7	12
3	0	2	2	2

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

	X1	X2	X3	b
1	1	-1	-2	-2
2	0	1	3.5	6
3	0	2	2	2

Multiply the 2nd row by -1

	X1	X2	X3	b
1	1	-1	-2	-2
2	0	-1	-3.5	-6
3	0	2	2	2

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

	X1	X2	X3	b
1	1	0	1.5	4
2	0	1	3.5	6
3	0	2	2	2

Multiply the 2nd row by 2

	X1	X2	X3	b
1	1	0	1.5	4
2	0	2	7	12
3	0	2	2	2

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

	X1	X2	X3	b
1	1	0	1.5	4
2	0	1	3.5	6
3	0	0	-5	-10

Make the pivot in the 3rd column by dividing the 3rd row by -5

	X1	X2	X3	b
1	1	0	1.5	4
2	0	1	3.5	6
3	0	0	1	2

Multiply the 3rd row by 1.5

	X1	X2	X3	b
1	1	0	1.5	4
2	0	1	3.5	6
3	0	0	1.5	3

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

	X1	X2	X3	b
1	1	0	0	1
2	0	1	3.5	6
3	0	0	1	2

Multiply the 3rd row by 3.5

	X1	X2	X3	b
1	1	0	0	1
2	0	1	3.5	6
3	0	0	3.5	7

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

	X1	X2	X3	b
1	1	0	0	1
2	0	1	0	-1
3	0	0	1	2

Solution set:

x1 = 1

x2 = -1

x3 = 2