Gauss Example ¶

We will solve the following system using the GAUSS method.

The corresponding matrix is ¶

\[ \begin{split}\ \begin{pmatrix} \ 0 & -1 & 2 & 13 & 5 \\ \ 1 & -2 & 3 & 17 & 4 \\ \ -1 & 3 & -3 & -20 & -1 \end{pmatrix}\end{split} \]

Step 1 ¶

Operation: $L_1 \iff L_2$

\[ \begin{split}\ \begin{pmatrix} \ 1 & -2 & 3 & 17 & 4 \\ \ 0 & -1 & 2 & 13 & 5 \\ \ -1 & 3 & -3 & -20 & -1 \end{pmatrix}\end{split} \]

Step 2 ¶

Operation: $L_3 = L_3 + L_1$

\[ \begin{split}\ \begin{pmatrix} \ 1 & -2 & 3 & 17 & 4 \\ \ 0 & -1 & 2 & 13 & 5 \\ \ 0 & 1 & 0 & -3 & 3 \end{pmatrix}\end{split} \]

Step 3 ¶

Operation: $L_3 = L_3 + L_2$

\[ \begin{split}\ \begin{pmatrix} \ 1 & -2 & 3 & 17 & 4 \\ \ 0 & -1 & 2 & 13 & 5 \\ \ 0 & 0 & 2 & 10 & 8 \end{pmatrix}\end{split} \]

At this point, the matrix is in the row echelon form. We could write back the system

\[ \begin{split}\left \{ \begin{array}{r c l} x_1 - 2x_2 + 3x_3 + 17x_4 = 4\\ - x_2 + 2x_3 + 13x_4 = 5\\ 2x_3 + 10x_4 = 8\\ \end{array} \right .\end{split} \]

But let's continue until we get the reduced row echelon form!

Step 4 ¶

Operation: $L_3 = L_3/2$

\[ \begin{split}\ \begin{pmatrix} \ 1 & -2 & 3 & 17 & 4 \\ \ 0 & -1 & 2 & 13 & 5 \\ \ 0 & 0 & 1 & 5 & 4 \end{pmatrix}\end{split} \]

Step 5 ¶

Operation: $L_3 = L_1/-1$

\[ \begin{split}\ \begin{pmatrix} \ 1 & -2 & 3 & 17 & 4 \\ \ 0 & 1 & -2 & -13 & -5 \\ \ 0 & 0 & 1 & 5 & 4 \end{pmatrix}\end{split} \]

Step 6 ¶

Operation: $L_1 = L_1 - 3 L_3$ and $L_2 = L_2 + 2 L_3$

\[ \begin{split}\ \begin{pmatrix} \ 1 & -2 & 0 & 2 & -8 \\ \ 0 & 1 & 0 & -3 & 3 \\ \ 0 & 0 & 1 & 5 & 4 \end{pmatrix}\end{split} \]

Step 7 ¶

Operation: $L_1 = L_1 + 2 L_2$

\[ \begin{split}\ \begin{pmatrix} \ 1 & 0 & 0 & -4 & -2 \\ \ 0 & 1 & 0 & -3 & 3 \\ \ 0 & 0 & 1 & 5 & 4 \end{pmatrix}\end{split} \]

Result ¶

We are converting it back to a system

\[ \begin{split}\left \{ \begin{array}{r c l} x_1 - 4x_4 = -2\\ x_2 - 3x_4 = 3\\ x_3 + 5x_4 = 4\\ \end{array} \right .\end{split} \]

Giving us

\[ \begin{split}\left \{ \begin{array}{r c l} x_1 = -2 + 4x_4\\ x_2 = 3 + 3x_4\\ x_3 = 4 - 5x_4\\ \end{array} \right .\end{split} \]