Question

How do you determine whether a matrix is in echelon form, reduced echelon form or not in echelon form?

Answer 1

Hint : We have to first define the conditions which change a regular matrix into echelon form and reduced echelon form. The row operations have to be done in a particular process to achieve that. From the conditions followed by the matrix we determine whether a matrix is in echelon form, reduced echelon form or not in echelon form.

Complete step by step solution:
There are two types of echelon forms which are achieved by row operations on a matrix. These are row echelon form, reduced echelon form.
First, we describe the forms and their conditions.
A row echelon form is achieved when the first non-zero element in each row is 1. Each leading entry is in a column to the right of the leading entry in the previous row. Rows with all zero elements will be below the rows having at least one non-zero element.
Example of this form is $\left[ \begin{matrix} 1 & 2 & 3 & 4 \\\ 0 & 0 & 1 & 3 \\\ 0 & 0 & 0 & 1 \\\ 0 & 0 & 0 & 0 \\\ \end{matrix} \right]$.
In reduced echelon form it follows all the conditions followed by the row echelon form in addition with the leading entry in each row is the only non-zero entry in its column.
The example being $\left[ \begin{matrix} 1 & 2 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1 \\\ 0 & 0 & 0 & 0 \\\ \end{matrix} \right]$
If a matrix after row operations do not follow these rules properly, then we can say that the matrix is not in echelon form.
The example being $\left[ \begin{matrix} 1 & 2 & 3 & 4 \\\ 1 & 0 & 2 & 3 \\\ 0 & 0 & 0 & 1 \\\ 0 & 0 & 0 & 1 \\\ \end{matrix} \right]$.

Note : The matrix operations and steps for the conversion have to be row operations. The processes can be addition, subtraction and multiplied binary operations between the rows.