Matrix Row Switching

Row Switching

Row switching is probably the simplest row operation. All you do is take one row and switch it with a second row. It is denoted with $R_i\longleftrightarrow R_j$ which just means switch the $i^{th}$ row with the $j^{th}$ row.

Example

Let $A=\begin{bmatrix}\color{blue}{a}&\color{blue}{b}&\color{blue}{c}\\d&e&f\\\color{red}g&\color{red}h&\color{red}i\end{bmatrix}$

Now say we want to apply the row operation: $R_1\longleftrightarrow R_3$

The matrix now becomes: $\begin{bmatrix}\color{red}{g}&\color{red}{h}&\color{red}{i}\\d&e&f\\\color{blue}{a}&\color{blue}{b}&\color{blue}{c}\end{bmatrix}$

Row Switching

One of the most common types of matrix elementary row operations is called row switching. Here, we basically switch two rows in a matrix. This is the first of the matrix basic operations.

Let's look at the following elementary matrix:

Example

If we wanted to apply the row switching technique to the following matrix,

$$\begin{bmatrix} 2&4&6 \\ 3&5&7 \\ \end{bmatrix}$$

We could say that $R_1 \rightarrow R_2$:

$$\begin{bmatrix} 3&5&7 \\ 2&4&6 \\ \end{bmatrix}$$

We switched the first and second rows in the matrix!

The Tricky Part

The tricky part of matrix row switch comes up most commonly with the identity matrix. When you're dealing with these, you want to make sure you keep track of the elements in a row.

Looking Further

In the next few lessons, we will cover other types of matrix operations, such as matrix row multiplication and matrix row addition. With multiplication, you are multiplying a row by a number, and with addition, you are adding values.

You may use these matrix row operations with advanced linear systems, so take the time to get these foundations right!

(Video) Matrix Row Operations: Matrix Row Switching

by Anusha Rahman

In this video, Anusha Rahman goes over what matrix row switching is and walks you through an example.

Summary

Matrix row switching allows us to switch rows within a matrix.

If given Matrix $A$, $\begin{bmatrix} 1&2&3 \\ 4&5&6 \\ 7&8&9 \\ \end{bmatrix}$, we may want to switch rows $1$ and $3$, $R_1 \rightarrow R_3$. To do that, we will literally swap the first and third rows of Matrix $A$. That means our new matrix looks like,

$\begin{bmatrix} 7&8&9 \\ 4&5&6 \\ 1&2&3 \\ \end{bmatrix}$