Precalculus (6th Edition) Blitzer

The basic row operations are: Interchanging any two rows. Multiplying any row by a non-zero real number. Adding one row to any other row. Examples: Interchange any two rows. In this operation, a row can be interchanged with any other row. Consider a matrix, $\left[ \begin{matrix} 1 & -3 & 5 \\ 1 & -2 & 7 \\ 2 & 1 & 4 \\ \end{matrix} \right]$ Interchange the first row and second row; that is ${{R}_{1}}\leftrightarrow {{R}_{2}}$ The resulting matrix is, $\left[ \begin{matrix} 1 & -2 & 7 \\ 1 & -3 & 5 \\ 2 & 1 & 4 \\ \end{matrix} \right]$ Multiply any row by a non-zero real number. In this operation, a row can be multiplied by any non-zero real number. Consider the matrix, $\left[ \begin{matrix} 1 & -3 & 5 \\ 1 & -2 & 7 \\ 2 & 1 & 4 \\ \end{matrix} \right]$ Multiply the first row by $2$; that is ${{R}_{1}}\to 2{{R}_{1}}$ The resulting matrix is, $\left[ \begin{matrix} 2 & -6 & 10 \\ 1 & -2 & 7 \\ 2 & 1 & 4 \\ \end{matrix} \right]$ Add one row to any other row. In this operation, any row can be added to another row. Consider the matrix, $\left[ \begin{matrix} 1 & -3 & 5 \\ 1 & -2 & 7 \\ 2 & 1 & 4 \\ \end{matrix} \right]$ Add the second row to the first row; that is ${{R}_{1}}\to {{R}_{1}}+{{R}_{2}}$ The resulting matrix is, $\left[ \begin{matrix} 2 & -1 & 12 \\ 1 & -2 & 7 \\ 2 & 1 & 4 \\ \end{matrix} \right]$