1. The problem is to understand the three types of pivot operations used in matrix row operations. 2. The three pivot operations are: - Interchange two rows: $R_i \leftrightarrow R_j$ - Multiply a row by a nonzero scalar: $kR_i$, where $k \neq 0$ - Add a multiple of one row to another: $R_i \to kR_j + R_i$ 3. These operations are fundamental in methods like Gaussian elimination to solve systems of linear equations. 4. Important rules: - Interchanging rows swaps their positions. - Multiplying a row by a nonzero scalar scales all elements in that row. - Adding a multiple of one row to another replaces the target row with the sum. 5. These operations preserve the solution set of the system represented by the matrix. 6. Understanding these helps in performing row reduction to echelon forms. Final answer: The three pivot operations are $R_i \leftrightarrow R_j$, $kR_i$ with $k \neq 0$, and $R_i \to kR_j + R_i$.