Gaussian elimination method
\[\Large
\begin{bmatrix}
a_{11} + a_{12}+ a_{13}…….. + a_{1n} \\
a_{21} + a_{22}+ a_{23}…….. + a_{2n} \\
⋮~~~~~~~⋮~~~~~~⋮~~~~~~~~~~~~~~⋮ \\
a_{n1} + a_{n2}+ a_{n3}…….. + a_{nn}
\end{bmatrix} ·
\begin{bmatrix}
x_{1} \\
x_{2} \\
⋮ \\
x_{n}
\end{bmatrix} =
\begin{bmatrix}
b_{1} \\
b_{2} \\
⋮ \\
b_{n}
\end{bmatrix}
~~~~i.e~ A·x = b
\]
All the information for solving the set of equations is provided by the matrix of coefficients A and the column matrix b. If we write the elements of b within the matrix A, we obtain the augmented matrix B of the given set of equations.
\[\Large
\begin{bmatrix}
a_{11} + a_{12}+ a_{13}…….. + a_{1n} ~|~~ b_{1}\\
a_{21} + a_{22}+ a_{23}…….. + a_{2n} ~|~~b_{2}\\
⋮~~~~~~~⋮~~~~~~⋮~~~~~~~~~~~~~~⋮~~~~~~~~~~~~|\\
a_{n1} + a_{n2}+ a_{n3}…….. + a_{nn} ~|~~b_{n}
\end{bmatrix}
\]
We then eliminate the elements other than a11 from the first column by subtracting a21/a11 times the first row from the second row and a31=a11 times the first row from the third row, etc.
This gives a new matrix of the form:
\[\Large
\begin{bmatrix}
a_{11} + a_{12}+ a_{13}…….. + a_{1n} ~|~~ b_{1}\\
0 + c_{22}+ c_{23}…….. + c_{2n} ~~~~~|~~b_{2}\\
⋮~~~~~~~⋮~~~~~~⋮~~~~~~~~~~~~~~⋮~~~~~~~~~~~~|\\
0 + c_{n2}+ c_{n3}…….. + c_{nn} ~~~~~|~~d_{n}
\end{bmatrix}
\]
The process is then repeated to eliminate ci2 from the third and subsequent rows.
A specific example will explain the method, so move on to the next page