Gaussian Elimination/Examples/Arbitrary Matrix 3

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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.

$\blacksquare$


Sources

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