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 simplify it or solve 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 to another. These operations are fundamental in methods like Gaussian elimination, which is used to find solutions to linear systems.

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. Achieving REF is crucial for solving linear equations, as it simplifies the process of back substitution to find variable values.

Scalar multiplication involves multiplying each entry of a matrix by a constant (scalar). This operation is essential in row operations, as it allows for the adjustment of a row's values without changing the relationships between the equations represented by the matrix. For example, multiplying the first row by 1/2, as indicated in the question, scales down the values, making calculations easier while maintaining the row's proportionality.