Why do I get differnt row reduced echelon form?

Question

As far as I know ,one matrix has a unique row reduced echelon form. But why do I get the different echelon form if I change the last row operation of the first picture? What have I done wrong ? What's the exact rule here so that I can get the same result all the time?

Answer 1

RREF is unique.

Review the definition of elementary row operations.

• We can multiply a row by non-zero number.
• We can swap two rows
• We can perform the operation $R_{\color{red}{i}} = R_\color{red}i + cR_\color{blue}j$

Notice that row operations doesn't include the operation $R_{\color{red}{i}} = R_\color{blue}j + cR_\color{red}i$