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
[Math] Is there no difference between upper triangular matrix and echelon matrix(row echelon matrix)
linear algebramatrices
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Best Answer
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.