Elementary Transformation

The six elementary transformations. An elementary transformation of a matrix is one of six operations -- three acting on rows, and three (their mirror images) acting on columns:

1. Interchange of any two rows or any two columns. If the $i$-th row and the $j$-th row of a matrix are interchanged, the original matrix is transformed into a new matrix. This is written $R_i \leftrightarrow R_j$ (or $R_{ij}$ for short); the column version is $C_i \leftrightarrow C_j$ (or $C_{ij}$). For example, if $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ then $R_1\leftrightarrow R_2$ gives the new matrix $\begin{bmatrix}3&4\\1&2\end{bmatrix}$, and $C_1\leftrightarrow C_2$ gives the new matrix $\begin{bmatrix}2&1\\4&3\end{bmatrix}$. Note carefully that $A \neq \begin{bmatrix}3&4\\1&2\end{bmatrix}$ and $A \neq \begin{bmatrix}2&1\\4&3\end{bmatrix}$ -- the transformed matrix is a genuinely different matrix, not equal to $A$. Instead the relationship is written using the symbol $\sim$ (read "is equivalent to"): $A \sim \begin{bmatrix}3&4\\1&2\end{bmatrix}$ and $A \sim \begin{bmatrix}2&1\\4&3\end{bmatrix}$.
2. Multiplication of the elements of any row or column by a non-zero scalar. If $k$ is a non-zero scalar, multiplying every element of row $R_i$ by $k$ is written $R_i \to kR_i$ (and $C_l \to kC_l$ for a column). For example, if $A = \begin{bmatrix}0&2\\3&4\end{bmatrix}$ then $R_2 \to 4R_2$ gives $A \sim \begin{bmatrix}0&2\\12&16\end{bmatrix}$; and $C_1 \to -3C_1$ on the same $A$ gives $A \sim \begin{bmatrix}0&2\\-9&4\end{bmatrix}$. In both cases the result is only equivalent to $A$, never equal to it.
3. Adding the scalar multiples of all the elements of any row (column) to the corresponding elements of any other row (column). If $k$ is a non-zero scalar, adding the $k$-multiples of the elements of $R_i$ (or $C_i$) to the elements of $R_j$ (or $C_j$) is written $R_j \to R_j + kR_i$, $C_j \to C_j + kC_i$. For example, if $A = \begin{bmatrix}-1&4\\2&5\end{bmatrix}$ and $k=2$, then $R_1 \to R_1 + 2R_2$ gives $A \sim \begin{bmatrix}-1+2(2)&4+2(5)\\2&5\end{bmatrix} = \begin{bmatrix}3&14\\2&5\end{bmatrix}$. Two bookkeeping notes on transformation (c):

1. After $R_j \to R_j + kR_i$, the row $R_i$ that supplied the multiple is left completely unchanged in the new matrix -- only $R_j$ is overwritten. The same holds for columns: after $C_j \to C_j + kC_i$, column $C_i$ is unchanged.
2. After any elementary transformation, the matrix obtained is said to be equivalent to the original matrix (never equal, except in the trivial case where the transformation happens to have no effect). …