Elementary Transformations of a Matrix

An elementary transformation (or elementary operation) is one of three simple changes applied to the rows or columns of a matrix. They are the engine behind two methods later in the chapter — finding an inverse, and solving a system of equations by reduction — so it is worth learning the standard notation for them.

The three elementary row transformations are:

1. Interchange of two rows, written $R_i\leftrightarrow R_j$.
2. Multiplication of a row by a non-zero scalar, written $R_i\to kR_i$ (with $k\neq 0$).
3. Addition to one row of a scalar multiple of another row, written $R_i\to R_i+kR_j$.

Exactly the same three operations applied to columns ($C_i\leftrightarrow C_j$, $C_i\to kC_i$, $C_i\to C_i+kC_j$) are the elementary column transformations.

The purpose of these operations is to simplify a matrix — typically to create zeros in chosen positions — without changing the essential problem being solved. When they are used to find an inverse, only row operations (or only column operations) must be used throughout a single computation; mixing rows and columns in the same inversion is a common and fatal error. …