# Is there no difference between upper triangular matrix and echelon matrix(row echelon matrix)? 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

• A triangular is a square matrix while an echelon matrix is a rectangular matrix, it is more general. Mar 30, 2016 at 17:57
• Any $m \times n$ matrix can be in row-echelon form. There is a requirement for a triangular matrix to be square. That is the difference. Mar 30, 2016 at 17:58
• Try to find a (singular) upper triangular matrix that is not in echelon form. Mar 30, 2016 at 17:59
• If we define $A$ be $A=(a_{ij})$ . $a_{ij}=0, if i>j$ be a an upper triangualr matrix. then there is no difference.
– user464147
Jan 3, 2019 at 16:21

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: Can square matrices be in row echelon form? Shouldn't row echelon matrices be rectangular by definition? Or are the quotes implying that row echelon is for rectangles what triangular is for squares? Mar 27, 2017 at 0:50