Elementary Row Operations on Matrices

This page is intended to be a part of the Numerical Analysis section of Math Online. Similar topics can also be found in the Linear Algebra section of the site.

# Elementary Row Operations on Matrices

Suppose that we have an $m \times n$ matrix $A$. There are three operations that we can perform directly on the matrix known as Elementary Row Operations which we define below.

 Definition: Let $A$ be an $m \times n$ matrix. Then the Elementary Row Operations that can be performed on $A$ are: a) The addition of a scalar multiple $k$ of row $R_b$ to row $R_a$, that is $R_a + kR_b \to R_a$. b) The multiplication of a row $R_a$ by a nonzero scalar $k$, that is $kR_a \to R_a$. c) The interchange of $R_a$ and $R_b$, that is $R_a \leftrightarrow R_b$.

Here, $R_a$ and $R_b$ represent arbitrary rows of $A$, and so $a, b = 1, 2, ..., m$. The notation $R_a + kR_b \to R_a$ should be read as "Row $a$ plus $k$ times row $b$ replaces row $a$. The notation $kR_a \to R_a$ should be read as "$k$ times row $a$ replaces row $a$. Lastly, the notation $R_a \leftrightarrow R_b$ should be read as "row $a$ is interchanged with row $b$".

For example, consider the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$. We will exhibit each operation successively on this matrix.

First let's take the first row and add $5$ times the second row to it. This would be denoted $R_1 + 5R_2 \to R_1$ and we get the following matrix

(1)
\begin{align} \quad \begin{bmatrix} 1 + 5(4) & 2 + 5(5) & 3 + 5(6) \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} 21 & 27 & 33 \\ 4 & 5 & 6 \end{bmatrix} \end{align}

Now let's take this matrix and multiply the second row by $4$. This would be denoted $4R_2 \to R_2$ and we get the following matrix:

(2)
\begin{align} \quad \begin{bmatrix} 21 & 27 & 33 \\ 4 \cdot 4 & 5 \cdot 4& 6 \cdot 4 \end{bmatrix} = \begin{bmatrix} 21 & 27 & 33 \\ 16 & 20 & 24 \end{bmatrix} \end{align}

Lastly, let's take this matrix and interchange row one with row two. This would be denoted $R_1 \leftrightarrow R_2$ and we get the following matrix:

(3)
\begin{bmatrix} 16 & 20 & 24 \\ 21 & 27 & 33 \end{bmatrix}

We will later see the important in these elementary row operations when dealing with solving systems of linear equations.