Answer
Interchange any two rows; multiply any row by a non-zero real number and add one row to any other row.
Work Step by Step
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,
$\left[ \begin{matrix}
2 & -1 & 12 \\
1 & -2 & 7 \\
2 & 1 & 4 \\
\end{matrix} \right]$
