Enguneering maths, Echelon form

Question

can a matrix have more than one echelon form??

Answer 1

The reduced row echelon form is unique. The are many paths to this form, but the destination is unique.

Source: stackoverflow

The reduced row echelon form of a matrix may be computed by Gauss–Jordan elimination. Unlike the row echelon form, a matrix's reduced row echelon form is unique and does not depend on the algorithm used to compute it.

Source: Wikipedia

So yes a matrix can have multiple echelon forms but a reduced row echelon form will be unique

For example: If a matrix has $A$ has the following echelon form $\begin{bmatrix}
1 & 2 & 3\\
0 & 1 & 2\\
0 & 0 & 1
\end{bmatrix}$

then other possible echelon forms are $\begin{bmatrix}
1 & 2 & 3\\
0 & 1 & 2\\
0 & 0 & 2
\end{bmatrix}$    ,   $\begin{bmatrix}
1 & 2 & 3\\
0 & 2 & 4\\
0 & 0 & 1
\end{bmatrix}$, …..