4.6 Elementary Row Operations

4.6 Elementary Row Operations

Elementary operations on equations (swap, scale by nonzero $k$, add multiple of one equation to another) correspond to row operations on matrices:

• $R_i \leftrightarrow R_j$
• $kR_i \rightarrow R'_i$ ($k \neq 0$)
• $R_i + kR_j \rightarrow R'_i$

If $B$ is obtained from $A$ by finitely many row operations, write $A \cong B$ (equivalent matrices).

Upper triangular: $a_{ij} = 0$ for all $i > j$.

Lower triangular: $a_{ij} = 0$ for all $i < j$.

Triangular: upper or lower. Diagonal matrices are both.