Question

Explain what the Gaussian elimination does, by column picture, to a linear system with 3 unknowns and 3 equations.

Answer 1

The best way to explain is by using an example. Let AX = b be the matrix representation of the following system of linear equations:

x+y+z = 1…(1)

x-y+z = 4…(2)

x+2y+4z = 7…(3).

Here, A is the coefficient matrix

1	1	1
1	-1	1
1	2	4

X = (x,y,z)^T and b = (1,4,7)^T. The augmented matrix is M(say) =

1	1	1	1
1	-1	1	4
1	2	4	7

The Gaussian elimination reduces the system to the reduced echelon form using elementary row operations as under:

Add -1 times the 1st row to the 2nd row

Add -1 times the 1st row to the 3rd row

Multiply the 2nd row by -1/2

Add -1 times the 2nd row to the 3rd row

Multiply the 3rd row by 1/3

Add -1 times the 3rd row to the 1st row

Add -1 times the 2nd row to the 1st row

Then the reduced row echelon form of M is

1	0	0	0
0	1	0	-3/2
0	0	1	5/2

It may be observed that every column of the coefficient matrix has a pivot position. The Gaussian elimination uses elementary row operations to create pivot position in every column of the coefficient matrix.

The solution is x = 0, y = -3/2 and z = 5/2.