> [!summary] A matrix in row echelon form is not a unique matrix. >[!info]+ Read Time **⏱ 1 min** # Definition Row echelon form (REF) of a [[Matrix Notation|matrix]] is a matrix that is in a non-unique form. Meaning you could do some type of [[Elementary Row Operations (ERO)|row operation]] and end up with a different type of matrix in the row echelon form. For a matrix to be in row echelon form, the [[Matrix Notation|coefficient matrix]] must have the following properties: - If a row does not consist entirely of zero, then the first non-zero number must be a 1 - All zero rows are at the bottom of the matrix - In any two successive rows that do not consist entirely of zeros, the [[Leading 1s|leading 1]] in the row occurs farther to the right than the [[Leading 1s|leading 1 ]] on the row above it # Examples > [!example] Examples of matrices in row echelon form > $ \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 5 \end{bmatrix} > $ > $ \begin{bmatrix} 1 & 0 & -2 & 3 \\ 0 & 1 & 4 & -1 \\ 0 & 0 & 2 & 5 \end{bmatrix} > $ > $ \begin{bmatrix} 1 & 0 & 0 & 0 \\ -1 & 2 & 0 & 0 \\ 2 & 3 & 1 & 0 \\ 0 & -1 & 5 & 4 \end{bmatrix} > $ > [!example] Examples of matrices **not** in row echelon form > $ \begin{bmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \\ 0 & 2 & 1 \end{bmatrix} > $ > >$ \begin{bmatrix} 0 & 1 & 2 & 3 \\ 2 & 0 & 4 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix} > $ > > $ \begin{bmatrix} 3 & 5 & -2 \\ 6 & 1 & 0 \\ 1 & 4 & 4 \end{bmatrix} > $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/oXMPQ- 6YnGA?si=KovUfjJ9sDJ44oTz" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>