Gaussian Elimination/Examples/Arbitrary Matrix 6

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


 * $\mathbf A = \begin {bmatrix}

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

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


 * $\mathbf E = \begin {bmatrix}

1 & 0 & 1 + i & 1    \\ 0 & 1 & \dfrac {1 + i} 2 & \dfrac {1 - i} 2 \\ 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 $\mathbf A_k$.

$e_1 := r_2 \leftrightarrow r_1$

Hence:
 * $\mathbf A_1 = \begin {bmatrix}

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

$e_2 := r_3 \to r_3 - r_1$

$e_3 := r_3 \to r_3 + r_2$


 * $\mathbf A_3 = \begin {bmatrix}

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

$e_4 := r_2 \to r_2 - i r_1$


 * $\mathbf A_4 = \begin {bmatrix}

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

$e_5 := r_1 \to r_1 + r_3$


 * $\mathbf A_5 = \begin {bmatrix}

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

$e_6 := r_3 \to \dfrac {r_3} 2$


 * $\mathbf A_6 = \begin {bmatrix}

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

$e_7 := r_2 \to r_2 - \paren {1 + i} r_3$


 * $\mathbf A_7 = \begin {bmatrix}

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

$e_8 := r_2 \leftrightarrow r_3$


 * $\mathbf A_8 = \begin {bmatrix}

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

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