◀ ▲ ▶Branches / Algebra / Example: The Gaussian Method in Practice
The following example will demonstrate the Gaussian method by example.
Example: The Gaussian Method in Practice
(related to Section: Solving General Systems Of Linear Equations - Gaussian Method)
Example
We want to solve the system of linear equations with three unknowns
$$\begin{array}{rcr}
x_1 -3x_2 +2x_3&=&1\\
5x_1 + 4x_2 -3x_3&=&4\\
2x_1 -8x_2 +4x_3&=& -2\\
\end{array}\quad\quad( * )$$
This system has the following extended coefficient matrix:
$$A|\beta:=
\left(\begin{array}{rrr|r}
1&-3&2&1\\
5&4&-3&4\\
2&-8&4&-2\\
\end{array}\right)$$
In the following, we use SageMath. You will have to click the evaluate buttons to see the results.
print("Original extended matrix:\n")
A=matrix(QQ,[[1,-3,2,1],[5,4,-3,4],[2,-8,4,-2]])
print(A)
print("")
print("STEP 1: Bring the matrix to the upper-triangular form")
print("\nAdding the -5-fold multiple of the first row to the second:\n")
A1=A
A1[1]=A1[1]-5*A1[0]
print(A1)
print("\nAdding the -2-fold multiple of the first row to the third:\n")
A1[2]=A1[2]-2*A1[0]
print(A1)
print("\nAdding the 2/19-fold multiple of the second row to the third:\n")
A1[2]=A1[2]+2/19*A1[1]
print(A1)
The resulting upper-triangular matrix is
$$A|\beta:=
\left(\begin{array}{rrr|r}
1&-3&2&1\\
0&19&-13&-1\\
0&0&-\frac{26}{19}&-\frac{78}{19}\\
\end{array}\right)$$
Now we can use the backward substitution to solve the system
print("STEP 2: Backward substitution:\n")
A=matrix(QQ,[[1,-3,2,1],[0,19,-13,-1],[0,0,-26/19,-78/19]])
print(A)
print("\nSetting x3=-78/19*(-19/26), substituting x3 in second row, setting x2=...etc....\n")
x3=A[2][3]/A[2][2]
x2=(A[1][3]-A[1][2]*x3)/A[1][1]
x1=(A[0][3]-A[0][2]*x3-A[0][1]*x2)/A[0][0]
print("x3=", x3)
print("x2=", x2)
print("x1=", x1)
Therefore, $x_1=1, x_2=2, x_3=3$ is the solution of the system $( * ).$
Thank you to the contributors under CC BY-SA 4.0! 
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References
Bibliography
- Knabner, P; Barth, W.: "Lineare Algebra - Grundlagen und Anwendungen", Springer Spektrum, 2013
