Elementary Row and Column Operations

Elementary row (column) operations on a matrix are exactly three moves:

1. Interchange any two rows (columns) of the matrix.
2. Scale a row (column) by a non-zero scalar.
3. Add to a row (column) a non-zero scalar multiple of another row (column).

Notation for row operations (column operations are written analogously):

• $R_i\leftrightarrow R_j$: interchange rows $i$ and $j$.
• $R_i\to\lambda R_i$ ($\lambda\ne0$): scale row $i$ by $\lambda$.
• $R_i\to R_i+\lambda R_j$ ($\lambda\ne0$): add $\lambda$ times row $j$ to row $i$.

Collectively, row and column operations are called elementary transformations.

Definition 1.4 (equivalent matrices). Two matrices $A$ and $B$ of the same order are equivalent, written $A\sim B$, if one can be obtained from the other by a sequence of elementary transformations.

Worked illustration. For $A=\begin{pmatrix}1&2&-1\\3&-4&2\\0&1&5\end{pmatrix}$, applying $R_2\to R_2-3R_1$ gives $B=\begin{pmatrix}1&2&-1\\0&-10&5\\0&1&5\end{pmatrix}$, and $A\sim B$. …