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.