My Findings

Gaussian elimination is a powerful and widely-used method for solving systems of linear equations. It is used across many fields, from engineering to economics, for finding the values of variables that satisfy a system of equations. Whether you’re working with data science models, electrical circuits, or optimization problems, understanding this technique is crucial.

Row Echelon Form

For each row in a matrix, if the row does not consist of only zeros, then the leftmost nonzero entry is called the pivot of that row.

If two leading entries are in the same column, then a row replacement operation can be used to make one of those entries zero.

Then by using the row swapping operation, one can always order the rows so that for every non-zero row, the leading entry is to the right of the leading entry of the row above.

The following matrix is in echelon form

Reduced Row Echelon Form (RREF)

A matrix is said to be in reduced row echelon form if

1. Leading entries are 1’s
2. Leading 1’s are to the right of those above
3. Rows with all zeroes are at the bottom
4. Each leading 1’s must be the only non-zero entry in its column

$\begin{pmatrix} 1 & 0 & 0 & -\frac{12}{25} \\ 0 & 1 & \frac{6}{5} & \frac{7}{5} \\ 0 & 0 & 1 & \frac{11}{10} \\ 0 & 0 & 0 & 0 \end{pmatrix}$

Row Echelon Form Example

Suppose the goal is to find and describe the set of solutions to the following system of linear equations

Then it’s associated augmented matrix is

Applying $R_2 + \frac{3}{2} R_1 \rightarrow R_2$ and $R_3 + R_1 \rightarrow R_3$

Applying $R_3 – 4 R_2 \rightarrow R_3$

Row Echelon Form Example

We will continue from the previous example

Applying $\frac{1}{2}R_1 \rightarrow R_1$

Applying $2R_2 \rightarrow R_2$ and $R_1 – \frac{1}{2} R_2 \rightarrow R_1$

Applying $R_2 – R_3 \rightarrow R_2$ and $R_1 + R_3 \rightarrow R_1$

Therefore the solution is $x = 2, y = 3, z = -1$

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 $A$:

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

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.

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

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 $A$:

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:

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 $E$

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

Row Scaling

For example, scaling row 1 by 3 would look like

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

Row Replacement

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

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