# Row Equivalence/Examples

## Examples of Row Equivalence

### Arbitrary Example $1$

Let $\mathbf A$ be the matrix defined as:

$\mathbf A := \begin {bmatrix} 1 & 0 & -1 & 1 \\ 2 & 1 & 0 & 1 \\ -1 & 1 & 0 & -2 \end {bmatrix}$

Let $\mathbf B$ be the matrix defined as:

$\mathbf B := \begin {bmatrix} 1 & 0 & -1 & 1 \\ -1 & 1 & 0 & 2 \\ 0 & 3 & 0 & 5 \end {bmatrix}$

Then $\mathbf A$ and $\mathbf B$ are row equivalent.

### Arbitrary Example $2$

Let $\mathbf A$ be the matrix defined as:

$\mathbf A := \begin {bmatrix} 1 & 0 & -1 \\ 2 & 1 & 0 \\ 1 & -1 & 1 \end {bmatrix}$

Let $\mathbf B$ be the matrix defined as:

$\mathbf B := \begin {bmatrix} 3 & -1 & 1 \\ 0 & 2 & 1 \\ 1 & -1 & 1 \end {bmatrix}$

Then $\mathbf A$ and $\mathbf B$ are row equivalent.

### Arbitrary Example $3$

Let $\mathbf A$ be the matrix defined as:

$\mathbf A := \begin {bmatrix} 1 & -1 & 1 & 2 \\ -2 & 3 & 0 & 1 \\ 1 & 0 & -1 & 3 \\ \end {bmatrix}$

Let $\mathbf B$ be the matrix defined as:

$\mathbf B := \begin {bmatrix} 0 & -1 & 2 & 3 \\ 1 & 2 & -1 & 0 \\ -2 & -5 & 4 & 3 \\ \end {bmatrix}$

Then $\mathbf A$ and $\mathbf B$ are not row equivalent.