Gaussian Elimination/Examples/Arbitrary Matrix 3

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


 * $\mathbf A = \begin {bmatrix}

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

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


 * $\mathbf E = \begin {bmatrix}

1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 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 $\mathbf A_k$.

$e_1 := r_2 \to r_2 - r_1$

$e_2 := r_3 \to r_3 - 2 r_1$

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

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

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

$e_4 := r_2 \to r_2 - r_4$


 * $\mathbf A_4 = \begin {bmatrix}

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

$e_5 := r_2 \to -r_2$


 * $\mathbf A_5 = \begin {bmatrix}

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

$e_6 := r_3 \to r_3 + 2 r_2$

$e_7 := r_4 \to r_4 + r_2$


 * $\mathbf A_7 = \begin {bmatrix}

1 & 1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 &  4 \\ 0 & 0 &  1 \\ \end {bmatrix}$

$e_8 := r_3 \to \dfrac {r_3} 4$


 * $\mathbf A_8 = \begin {bmatrix}

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

$e_9 := r_4 \to r_4 - r_3$


 * $\mathbf A_9 = \begin {bmatrix}

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

It may be noted that $\mathbf A_9$ is in echelon form.

It remains to convert it to reduced echelon form.

$e_{10} := r_1 \to r_1 - r_2$


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

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

$e_{11} := r_1 \to r_1 + r_3$


 * $\mathbf A_{11} = \begin {bmatrix}

1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end {bmatrix}$

and it is seen that $\mathbf A_{11}$ is the required reduced echelon form.