Gaussian Elimination/Examples/Arbitrary Matrix 2

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


 * $\mathbf A = \begin {bmatrix}

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

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


 * $\mathbf E = \begin {bmatrix}

1 & 0 & 0 & \dfrac 5 8 \\ 0 & 1 & 0 & -\dfrac 1 8 \\ 0 & 0 & 1 & \dfrac 1 8 \\ \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 $\mathbf A_k$.

$e_1 := r_2 \to r_2 - 2 r_1$

$e_2 := r_3 \to r_3 - r_1$

Hence:
 * $\mathbf A_2 = \begin {bmatrix}

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

$e_3 := r_2 \to \dfrac {r_2} 3$


 * $\mathbf A_3 = \begin {bmatrix}

1 & -1 &          2 &           1 \\ 0 &  1 & -\dfrac 5 3 & -\dfrac 1 3 \\ 0 & -1 & -1 & 0 \\ \end {bmatrix}$

$e_4 := r_3 \to r_3 + r_2$


 * $\mathbf A_4 = \begin {bmatrix}

1 & -1 &          2 &           1 \\ 0 &  1 & -\dfrac 5 3 & -\dfrac 1 3 \\ 0 & 0 & -\dfrac 8 3 &  -\dfrac 1 3 \\ \end {bmatrix}$

$e_5 := r_3 \to -\dfrac 3 8 r_3$


 * $\mathbf A_5 = \begin {bmatrix}

1 & -1 &          2 &           1 \\ 0 &  1 & -\dfrac 5 3 & -\dfrac 1 3 \\ 0 & 0 &           1 &  \dfrac 1 8 \\ \end {bmatrix}$

It may be noted that $\mathbf A_5$ is in echelon form.

It remains to convert it to reduced echelon form.

$e_6 := r_1 \to r_1 + r_2$


 * $\mathbf A_6 = \begin {bmatrix}

1 & 0 & \dfrac 1 3 &  \dfrac 2 3 \\ 0 & 1 & -\dfrac 5 3 & -\dfrac 1 3 \\ 0 & 0 &          1 &  \dfrac 1 8 \\ \end {bmatrix}$

$e_7 := r_2 \to r_2 + \dfrac 5 3 r_3$


 * $\mathbf A_7 = \begin {bmatrix}

1 & 0 & \dfrac 1 3 & \dfrac 2 3 \\ 0 & 1 &         0 & -\dfrac 1 8 \\ 0 & 0 &         1 &  \dfrac 1 8 \\ \end {bmatrix}$

$e_8 := r_1 \to r_1 - \dfrac 1 3 r_3$


 * $\mathbf A_8 = \begin {bmatrix}

1 & 0 & 0 & \dfrac 5 8 \\ 0 & 1 & 0 & -\dfrac 1 8 \\ 0 & 0 & 1 & \dfrac 1 8 \\ \end {bmatrix}$

and it is seen that $\mathbf A_8$ is the required reduced echelon form.