Row Equivalence/Examples/Arbitrary Example 1

Example of Row Equivalence
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.

Proof
$(1): \quad$ Use the elementary row operation on $\mathbf A$:
 * $e_1 := r_2 \to r_2 + 2 r_3$

to make $\mathbf A_1$:


 * $\mathbf A_1 := \begin {bmatrix} 1 & 0 & -1 & 1 \\ 0 & 3 & 0 & 5 \\ -1 & 1 & 0 & -2 \end {bmatrix}$

$(2): \quad$ Use the elementary row operation on $\mathbf A_1$:
 * $e_2 := r_2 \leftrightarrow r_3$

to make $\mathbf A_2$:


 * $\mathbf A_2 := \begin {bmatrix} 1 & 0 & -1 & 1 \\ -1 & 1 & 0 & -2 \\ 0 & 3 & 0 & 5 \end {bmatrix}$

which is $\mathbf B$.