Elementary Row Operations in Matrices

Elementary Row Operations (ERO)

Elementary row operations are fundamental tools in linear algebra used to manipulate matrices. These operations are critical in solving systems of linear equations, finding determinants, and performing matrix factorizations, such as Gaussian elimination.

In this article, we will explore the three main types of elementary row operations: row switching, row scaling, and row replacement, along with how to apply them using matrix multiplication.

There are three row operations:

Row Switching

The row switching operation swaps two rows of a matrix. This is represented as

$R_i \leftrightarrow R_j$

where rows $i$ and $j$ are exchanged. For example, if we have the following matrix $A$ :

A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}

And if we want to swap rows 1 and 2 ( $R_i \leftrightarrow R_j$ ), the new matrix will look like

A’ = \begin{pmatrix} 4 & 5 & 6 \\ 1 & 2 & 3 \\ 7 & 8 & 9 \end{pmatrix}

Row Scaling

Row scaling refers to multiplying all elements in a row by a scalar. This operation is represented as

$kR_i \rightarrow R_i$

For example, consider the matrix $A$ where $k \ne 0$ is a non-zero scalar and $i$ is the row index.

A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}

And if we want to scale first row $R_1$ by a factor of 3, the new matrix will look like:

A’ = \begin{pmatrix} 3 & 6 & 9 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}

Row Replacement

In row replacement, we add a scalar multiple of one row to another row. This operation is represented as where $i \ne j$ and $k$ is a scalar. For example, given the matrix $A$ :

A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}

If we want to replace the first row $R_1$ by adding 2 times the second row ( $R_2$ ), the operation is $R_1 + 2R_2 \rightarrow R_1$ , resulting in:

\begin{pmatrix} 9 & 12 & 15 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}

Matrix Multiplication

In order to apply the elementary row operations, we often represent these transformations using matrix multiplication. Here, we will show how each of the elementary row operations is represented as a multiplication of the original matrix by an elementary matrix.

Row Switching

For example, if we want to swap rows 1 and 2, we use the elementary matrix $E$

E = \begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix}

By multiplying $E$ by $A$ we get the result

\begin{align*} A’ &= E \times A \\ A’ &= \begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{pmatrix}\\ \begin{pmatrix} 4 & 5 & 6\\ 1 & 2 & 3\\ 7 & 8 & 9 \end{pmatrix} &= \begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{pmatrix}\\ \end{align*}

Row Scaling

For example, scaling row 1 by 3 would look like

E = \begin{pmatrix} 2 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}

By multiplying $E$ by $A$ we get the result

\begin{align*} A’ &= E \times A \\ A’ &= \begin{pmatrix} 2 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{pmatrix}\\ \begin{pmatrix} 2 & 4 & 6\\ 1 & 2 & 3\\ 7 & 8 & 9 \end{pmatrix} &= \begin{pmatrix} 2 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{pmatrix}\\ \end{align*}

Row Replacement

For example, replacing row 1 with $R_1 + 2R_2$ would look like

E = \begin{pmatrix} 1 & 0 & 0\\ 2 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}

By multiplying $E$ by $A$ we get the result

\begin{align*} A’ &= E \times A \\ A’ &= \begin{pmatrix} 1 & 0 & 0\\ 2 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{pmatrix}\\ \begin{pmatrix} 9 & 12 & 15\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{pmatrix} &= \begin{pmatrix} 1 & 0 & 0\\ 2 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{pmatrix}\\ \end{align*}

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