Elementary Operations of Matrix

What is it?

Apart from general mathematical operations, there are elementary operations used in Linear Algebra that are specific for working with Matrices.

They can be classified as elementary row operations for operations done on each row, or elementary column operations for operations done on each column.

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Interchange of rows and columns

Any two columns (or rows) of a matrix can be exchanged. If the $i^{th}$ and $j^{th}$ rows of a matrix are exchanged, it’s denoted as $R_i\leftrightarrow R_j$. For the columns, the notation is $C_i\leftrightarrow C_j$.

$$\begin{array}{rl}& A=\left[\begin{array}{ccc}1& 2& -3\\ 4& -5& 6\end{array}\right]\text{Original matrix }\\ & \\ & A=\left[\begin{array}{ccc}4& -5& 6\\ 1& 2& -3\end{array}\right]\text{Exchanged }{R}_1\leftrightarrow R_2\end{array}$$

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Multiplication by a non-zero number

Any row (or column) can be added with another row (or column), if multiplied by a non-zero number. Adding the $i^{th}$ row with the $j^{th}$ row, with the latter being multiplied by a non-zero number $k$, could be denoted as $R_i\to R_i+k{R}_j$. For columns the notation is $C_i\to C_i+k{C}_j$.

$$\begin{array}{rl}& A=\left[\begin{array}{ccc}1& 2& -3\\ 4& -5& 6\end{array}\right]\text{Original matrix }\\ & \\ & A=\left[\begin{array}{ccc}1& 2& -3\\ 8& 3& -7\end{array}\right]\text{Applied }{R}_2=R_2+4R_1\end{array}$$

It also can be applied to multiplication of a row, without the sum of another. While still needs to be a non-zero number, the notation is $R_i\to k{R}_i$ for rows and $C_i\to k{C}_i$ for columns.

$$\begin{array}{rl}& A=\left[\begin{array}{ccc}1& 2& -3\\ 4& -5& 6\end{array}\right]\text{Original matrix }\\ & \\ & A=\left[\begin{array}{ccc}3& 6& -9\\ 4& -5& 6\end{array}\right]\text{Applied }{R}_1=3R_1\end{array}$$