Is multiplication of a row by zero or a negative number allowed while performing a row operation?

Question

In matrices, while performing a row operation, am I allowed to multiply a row by zero or a negative number?

Answer 1

There are three types of row operations:

1. Multiplying a row by a constant $c\ne0$
2. Adding to a row a different row multiplied by a constant $d$
3. Swapping two rows

The third type is of no concern with your question.

Why the limitations on the first two types?

The idea is that row operations should be reversible, so when we perform a row operation, we can perform another one that “undoes” the effect of the previous one. The main reason is that such row operations don't change the solution set of a linear system (translated into matrix form), exactly because the operations are reversible.

Multiplying a row by $0$ is not reversible: this should be clear because doing this operation on all rows will yield the null matrix and we surely lose information. Thus is disallowed.

Adding to a row itself multiplied by $-1$ is the same as multiplying the row by $0$, so in general it is disallowed to add a row to itself multiplied by a scalar.

How do you reverse an operation of the first kind? By multiplying the same row by $c^{-1}$. What for an operation of the second kind? Say we add to row $i$ row $j$ multiplied by $d$; we reverse this by adding to row $i$ row $j$ multiplied by $-d$.

The constants $c$ and $d$ can be anything (but for the first type $c$ must be nonzero). Even $d=0$ is allowed in operations of the second kind: it is just doing nothing, which is of course reversed by doing nothing again.