Gaussian Elimination/Examples/Arbitrary Matrix 7

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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.

$\blacksquare$


Sources

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