Gaussian Elimination/Examples/Arbitrary Matrix 7

Example of Use of Gaussian Elimination
Let $\mathbf A$ denote the matrix:


 * $\mathbf A = \begin {bmatrix}

1 & 1 -   \sqrt 2 &  0           &        \sqrt 2 \\ \sqrt 2    & -3             &  1 + \sqrt 2 & -1 - 2 \sqrt 2 \\ -1 &       \sqrt 2 & -1           &  1             \\ \sqrt 2 - 2 & -2 + 4 \sqrt 2 & -2 - \sqrt 2 & 3 +   \sqrt 2 \\ \end {bmatrix}$

The reduced echelon form of $\mathbf A$ is:


 * $\mathbf E = \begin {bmatrix}

1 & 0 & 1 - \sqrt 2 &  0 \\ 0 & 1 & -1          &  0 \\ 0 & 0 &  0           &  1 \\ 0 & 0 &  0           &  0 \\ \end {bmatrix}$

Proof
In the following, $\sequence {e_n}_{n \mathop \ge 1}$ denotes the sequence of elementary row operations that are to be applied to $\mathbf A$.

The matrix that results from having applied $e_1$ to $e_k$ in order is denoted $\mathbf A_k$.

$e_1 := r_3 \to r_3 + r_1$

$e_2 := r_4 \to r_4 - r_2$

$e_3 := r_4 \to r_4 + 2 r_1$

Hence:
 * $\mathbf A_3 = \begin {bmatrix}

1 & 1 -   \sqrt 2 &  0             &        \sqrt 2 \\ \sqrt 2    & -3             &  1 +   \sqrt 2 & -1 - 2 \sqrt 2 \\ 0 & 1             & -1             &  1 +   \sqrt 2 \\ 0 & 3 + 2 \sqrt 2 & -3 - 2 \sqrt 2 &  4 + 5 \sqrt 2 \\ \end {bmatrix}$

$e_4 := r_2 \to r_2 - \sqrt 2 r_1$


 * $\mathbf A_4 = \begin {bmatrix}

1 & 1 -   \sqrt 2 &  0             &        \sqrt 2 \\ 0 & -1 -  \sqrt 2 &  1 +   \sqrt 2 & -3 - 2 \sqrt 2 \\ 0 & 1             & -1             &  1 +   \sqrt 2 \\ 0 & 3 + 2 \sqrt 2 & -3 - 2 \sqrt 2 &  4 + 5 \sqrt 2 \\ \end {bmatrix}$

$e_5 := r_2 \leftrightarrow r_3$


 * $\mathbf A_5 = \begin {bmatrix}

1 & 1 -   \sqrt 2 &  0             &        \sqrt 2 \\ 0 & 1             & -1             &  1 +   \sqrt 2 \\ 0 & -1 -  \sqrt 2 &  1 +   \sqrt 2 & -3 - 2 \sqrt 2 \\ 0 & 3 + 2 \sqrt 2 & -3 - 2 \sqrt 2 &  4 + 5 \sqrt 2 \\ \end {bmatrix}$

$e_6 := r_1 \to r_1 - r_3$

$e_7 := r_4 \to r_4 + 2 r_3$


 * $\mathbf A_7 = \begin {bmatrix}

1 & 2             & -1 -   \sqrt 2 &  3 + 3 \sqrt 2 \\ 0 & 1             & -1             &  1 +   \sqrt 2 \\ 0 & -1 -  \sqrt 2 &  1 +   \sqrt 2 & -3 - 2 \sqrt 2 \\ 0 & 1             & -1             & -2 +   \sqrt 2 \\ \end {bmatrix}$

$e_8 := r_1 \to r_1 - 2 r_2$

$e_9 := r_3 \to r_3 + \paren {1 + \sqrt 2} r_2$

$e_{10} := r_4 \to r_4 - r_2$


 * $\mathbf A_{10} = \begin {bmatrix}

1 & 0 & 1 -   \sqrt 2 &  1 +   \sqrt 2 \\ 0 & 1 & -1            &  1 +   \sqrt 2 \\ 0 & 0 & 0             &  0             \\ 0 & 0 &  0             & -3             \\ \end {bmatrix}$

$e_{11} := r_4 \to -\dfrac 1 3 r_4$

$e_{12} := r_3 \leftrightarrow r_4$


 * $\mathbf A_{12} = \begin {bmatrix}

1 & 0 & 1 - \sqrt 2 &  1 +   \sqrt 2 \\ 0 & 1 & -1          &  1 +   \sqrt 2 \\ 0 & 0 & 0           &  1             \\ 0 & 0 &  0           &  0             \\ \end {bmatrix}$

$e_{13} := r_1 \to r_1 - \paren {1 + \sqrt 2} r_3$

$e_{14} := r_2 \to r_2 - \paren {1 + \sqrt 2} r_3$


 * $\mathbf A_{12} = \begin {bmatrix}

1 & 0 & 1 - \sqrt 2 &  0 \\ 0 & 1 & -1          &  0 \\ 0 & 0 &  0           &  1 \\ 0 & 0 &  0           &  0 \\ \end {bmatrix}$

and it is seen that $\mathbf A_{12}$ is the required reduced echelon form.