In Exercises 13–18, perform each matrix row operation and write the new matrix.

Matrix row operations are techniques used to manipulate the rows of a matrix to achieve a desired form, typically for solving systems of equations. The three primary operations include swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row from another. These operations are fundamental in methods like Gaussian elimination and are essential for transforming matrices into row echelon form or reduced 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 simplifies the process of back substitution to find solutions to the equations represented by the matrix.

A linear combination involves creating a new vector (or row in a matrix) by adding together scalar multiples of existing vectors (or rows). In the context of matrix row operations, this means that one row can be replaced by a combination of itself and other rows, which is essential for simplifying matrices. Understanding linear combinations is key to grasping how row operations affect the overall structure and solutions of a matrix.