> [!summary] Elementary row operations are a set of operations for simple matrices > **Key Operations:** $R_{i} \leftrightarrow R_{i}$ $kR_{i} \quad | k\in \mathbb{R}$ $R_{i}\pm kR_{j}=R_{f}$ >[!info]+ Read Time **⏱ 2 mins** # Definition Elementary row operations are the simplest type of operations you can perform on a [[Matrix Notation|matrix]]. Since a [[Matrix Notation|matrix]] is just a notation for describing [[Linear Equations|linear equations]], any operations valid for [[Linear Equations|linear equations]] are valid for a [[Matrix Notation|matrix]]. > [!info] Operations > - Interchange two rows ($R_{i} \leftrightarrow R_{i}$) > - Multiply a row by a scalar amount ($kR_{i} \quad | k\in \mathbb{R}$) > - Replace a row by adding or subtracting other rows together ($R_{i}\pm kR_{j}=R_{f}$) # Examples > [!example] Solving the set of equations using linear equations and matrix notation > > $ \begin{array}{c} \text{Solve the set of linear equations using linear equations and matrix notation} \\ \begin{cases} x + 2y = 5, \\ 3x + 4y = 11 \end{cases} \\ \\ \\ \text{Matrix way} \\ \begin{bmatrix} 1 & 2 & 5 \\ 3 & 4 & 11 \end{bmatrix} \xrightarrow{R_2 - 3R_1} \begin{bmatrix} 1 & 2 & 5 \\ 0 & -2 & -4 \end{bmatrix} \xrightarrow{-\tfrac12 R_2} \begin{bmatrix} 1 & 2 & 5 \\ 0 & 1 & 2 \end{bmatrix} \xrightarrow{R_1 - 2R_2} \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix} \\ \\ \boxed{x = 1,\quad y = 2} \\ \\ \text{Linear equations way} \\ -3(x + 2y) = -15 \;\Longrightarrow\; -3x - 6y = -15, \\ (3x + 4y) + (-3x - 6y) = 11 - 15 \;\Longrightarrow\; -2y = -4 \;\Longrightarrow\; y = 2 \\ x + 2\cdot 2 = 5 \;\Longrightarrow\; x + 4 = 5 \;\Longrightarrow\; x = 1 \\ \\ \boxed{x = 1,\quad y = 2} \end{array} > $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/T2Gtt8WygiU?si=xoz4t_wgzWyNvISR" 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>