Gaussian Elimination/Examples

Examples of Use of Gaussian Elimination

Arbitrary Matrix $1$

Let $\mathbf A$ denote the matrix:

$\mathbf A = \begin {bmatrix}

0 & 0 & 5 & 35 & -24 & 1 \\ 0 & 2 & 1 & -1 & 1 & 0 \\ 0 & 3 & 2 & 2 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 5 & 3 & 1 & 0 & 1 \end {bmatrix}$

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

$\mathbf E = \begin {bmatrix}

0 & 1 & 0 & -4 & 0 & 26 \\ 0 & 0 & 1 & 7 & 0 & -43 \\ 0 & 0 & 0 & 0 & 1 & -9 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end {bmatrix}$

Arbitrary Matrix $2$

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}$

Arbitrary Matrix $3$

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}$

Arbitrary Matrix $4$

Let $\mathbf A$ denote the matrix:

$\mathbf A = \begin {bmatrix}

2 & 2 & 5 & 3 \\ 6 & 1 & 5 & 4 \\ 4 & -1 & 0 & 1 \\ 2 & 0 & 1 & 1 \\ \end {bmatrix}$

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

$\mathbf E = \begin {bmatrix}

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

Arbitrary Matrix $5$

Let $\mathbf A$ denote the matrix:

$\mathbf A = \begin {bmatrix}

0 & 1 & 1 &  1 &  2 & 2 \\

-1 & 4 & 3 & 3 & 4 & 7 \\

2 & 1 & 3 &  2 &  8 & 3 \\
3 & 1 & 4 & -1 &  4 & 0 \\
5 & 2 & 7 &  0 & 10 & 2 \\

\end {bmatrix}$

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

$\mathbf E = \begin {bmatrix}

1 & 0 & 1 &  0 &   2 &  0 \\
0 & 1 & 1 &  0 &   0 &  1 \\
0 & 0 & 0 &  1 &   2 &  1 \\
0 & 0 & 0 &  0 &   0 &  0 \\
0 & 0 & 0 &  0 &   0 &  0 \\

\end {bmatrix}$

Arbitrary Matrix $6$

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}$

Arbitrary Matrix $7$

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}$