TITLE: Is there no difference between upper triangular matrix and echelon matrix(row echelon matrix)?
QUESTION [8 upvotes]: Source: Linear Algebra with Applications Gareth Williams
I see no difference between upper triangular matrix and echelon matrix(row echelon matrix). Then are they the same?

Source: Linear Algebra with Applications David C. Lay

REPLY [13 votes]: To summarize the comments into an answer:
The matrix
$$\begin{pmatrix}1&2&3\\0&4&5\end{pmatrix} $$
is echelon, but not triangular (because not square).
The matrix
$$\begin{pmatrix}1&2&3\\0&0&4\\0&0&5\end{pmatrix} $$
is triangular, but not echelon (because the leading entry $5$ is not to the right of the leading entry $4$).
However, for non-singular square matrices, "row echelon" and "upper triangular" are equivalent.