In Exercises 1–2, perform each matrix row operation and write the new matrix.

Key Concepts

Matrix Row Operations

Matrix row operations are fundamental techniques used to manipulate matrices, primarily for solving systems of linear equations. The three types of row operations include swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row. These operations help in transforming a matrix into a simpler form, such as row echelon form or reduced row echelon form, which are essential for finding solutions.

Row Echelon Form

Row echelon form (REF) is a specific arrangement of a matrix where all non-zero rows are above any rows of all zeros, and the leading coefficient of each non-zero row (the first non-zero number from the left) is to the right of the leading coefficient of the previous row. This form is crucial for solving linear systems, as it allows for back substitution to find the values of the variables.

Matrix Notation

Matrix notation is a way of representing a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Each element in a matrix is identified by its position, typically denoted as a_{ij}, where 'i' is the row number and 'j' is the column number. Understanding matrix notation is essential for performing operations and communicating mathematical ideas clearly in linear algebra.