Q.$A = \begin{bmatrix} 1 & -1 & 3 \\ 2 & 1 & 0 \\ 3 & 3 & 1 \end{bmatrix}$, apply $C_3 + 2C_2$ and then $3R_3$. What do you conclude from ex. 5 and ex. 6?

An elementary transformation is one of six basic, reversible operations performed on the rows or columns of a matrix. There are three row transformations, each with a column counterpart: (a) interchange two rows/columns, written $R_i \leftrightarrow R_j$ or $C_i \leftrightarrow C_j$; (b) scalar multiplication of every entry of a row/column by a nonzero constant $k$, written $R_i \to kR_i$ or $C_i \to kC_i$; (c) scalar addition, where a multiple of one row/column is added entrywise to another, written $R_j \to R_j + kR_i$ or $C_j \to C_j + kC_i$. After transformation (a) or (c) the new matrix is generally not equal to the original, but the two are called equivalent matrices, written $A \sim B$, because each can be recovered from the other by a reverse elementary transformation. A key bookkeeping rule for transformation (c): in $R_j \to R_j + kR_i$, the row $R_i$ that supplies the multiple is left unchanged in the new matrix — only $R_j$ is overwritten (similarly for columns). Elementary transformations never change a matrix's order, and applying a row transformation never disturbs the effect of a previously-applied column transfor …