Is it possible to turn a matrix into its row-reduced echelon form by just using the row operations of the type the type $c R_i + R_j$ and $c R_i$?

Question

In the book of Linear Algebra by Cemal Koç, at page 62, it is asked that

Write $$\begin{matrix} 0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0 \end{matrix}$$ as a product of elementary row operations corresponding to the type $c R_i + R_j$ and $c R_i$

, and it can be done, we have tested.

However, I was wondering whether this is generally true ? I mean can we, for example, take a matrix and turn it into its row-reduced echelon form by just using the row operations of the type the type $c R_i + R_j$ and $c R_i$ ?

If so, how can we justify our result ? (And if not, again, how can we show that it is not possible to do so ?)

Answer 1

Yes you can. Proof:

Consider $a,b $ two rows. Now consider the following operations:

Set $a $ as $a+b$;

Set $b $ as $b-a$ then set $b $ as $-b $;

Set $a $ as $a-b $;

You effectively swapped rows with only the two row operations, so you can exclude explicit row swapping.

Of course for a real example there might be better operations to be done than this cheating swap.

Answer 2

Start with rows $R_1$, $R_2$. Then go to $R_1$, $R_1+R_2$. Now go to $R_1-(R_1+R_2),R_1+R_2$ which is $-R_2$, $R_1+R_2$. Then $-R_2$, $R_1+R_2+(-R_2)$ which is $-R_2$, $R_1$. Finally go to $R_2$, $R_1$.