Definition:Row Equivalence

Definition

Two matrices $\mathbf A = \sqbrk a_{m n}, \mathbf B = \sqbrk b_{m n}$ are row equivalent if one can be obtained from the other by a finite sequence of elementary row operations.

This relationship can be denoted $\mathbf A \sim \mathbf B$.

Examples

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.

Also see

• Row Equivalence is Equivalence Relation

• Definition:Column Equivalence

• Definition:Matrix Similarity

• Results about row equivalence can be found here.

Sources

• 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.2$ Elementary Row Operations on Matrices: Definition $1.3$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): row equivalence