# Category:Elementary Row Operations

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This category contains results about **Elementary Row Operations**.

Definitions specific to this category can be found in Definitions/Elementary Row Operations.

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$.

The **elementary row operations** on $\mathbf A$ are operations which act upon the rows of $\mathbf A$ as follows.

For some $i, j \in \closedint 1 m: i \ne j$:

\((\text {ERO} 1)\) | $:$ | \(\ds r_i \to \lambda r_i \) | For some $\lambda \in K_{\ne 0}$, multiply row $i$ by $\lambda$ | ||||||

\((\text {ERO} 2)\) | $:$ | \(\ds r_i \to r_i + \lambda r_j \) | For some $\lambda \in K$, add $\lambda$ times row $j$ to row $i$ | ||||||

\((\text {ERO} 3)\) | $:$ | \(\ds r_i \leftrightarrow r_j \) | Exchange rows $i$ and $j$ |

## Also see

## Subcategories

This category has the following 5 subcategories, out of 5 total.

### E

## Pages in category "Elementary Row Operations"

The following 11 pages are in this category, out of 11 total.

### E

- Effect of Elementary Row Operations on Determinant
- Effect of Sequence of Elementary Row Operations on Determinant
- Elementary Matrix corresponding to Elementary Row Operation
- Elementary Row Matrix for Inverse of Elementary Row Operation is Inverse
- Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations
- Elementary Row Operations as Matrix Multiplications
- Elementary Row Operations as Matrix Multiplications/Corollary
- Elementary Row Operations Commute with Matrix Multiplication
- Exchange of Rows as Sequence of Other Elementary Row Operations
- Existence of Inverse Elementary Row Operation