Theorem DRCMA: ambiguous statement

[darijgr]

What does "multiplying a row by the scalar $\alpha$ and then adding it to another row" mean in Theorem DRMCA? To me, it sounds like the following two-step procedure:

1. Multiply a row by the scalar $\alpha$;

2. Add it to another row.

Of course, this procedure changes both rows, and the determinant gets multiplied by $\alpha$ instead of remaining fixed.

I don't know how to reword it in a way that makes it clear that the first row is not actually changed, without making the result longer. Maybe say in a footnote that (like in Definition RO) you are "leaving the first row the same, but replacing the second row by the new values". Or just say "row operation of type 3"? (This would, of course, require you to introduce that notion.)

An example would certainly not hurt, too (not just an application of several theorems, but just an immediate example for DRMCA)...

[rbeezer]

Perhaps something like "use one row to change a second row by multiplying the first row by \alpha and adding it to the second row (leave the first row unchanged)"