> [!summary] A matrix in reduced row echelon form is a unique matrix. >[!info]+ Read Time **⏱ 2 mins** # Definition Reduced row echelon form (RREF) is a unique [[Matrix Notation|matrix]]. Meaning you can do some type of different [[Elementary Row Operations (ERO)|row operations]] and end up with the same matrix. A matrix is in reduced row echelon form if the [[Matrix Notation|coefficient matrix]] follows 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 1 on the row above it - **Each column containing a [[Leading 1s|leading 1]] has zeros in all its entries** The highlighted part of the properties is the difference between [[Row Echelon Form|row echelon form]] and reduced row echelon form # Examples > [!example] Example of matrices in reduced row echelon form > $ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} > $ > $ \begin{bmatrix} 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{bmatrix} > $ > $ \begin{bmatrix} 1 & 0 & 5 \\ 0 & 1 & -3 \end{bmatrix} > $ > [!example] Example of matrices **not** in reduced row echelon form > > $ \begin{bmatrix} 2 & 4 & 6 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \end{bmatrix} > $ > > $ \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 4 \\ 0 & 1 & 0 \end{bmatrix} > $ > > $ \begin{bmatrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} > $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/90YQb3Gajao?si=eWEIVKNPnWd86KTC" 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>