Rows that get zeroed out during elimination

[samh]

This is something that's bothered me for a very long time...

Say you're doing Gaussian elimination on some matrix.

Now suppose during the process, some row gets zeroed out, say row j. Now it makes perfect sense for me to say that row j was a linear combination of the previous rows.

But here's what I don't get. Suppose some row is a linear combination of rows above it. How does THAT imply that at some point during the elimination process THAT particular row will be zeroed out?

 

[NoMoreExams]

It won't necessarily if row k is a linear combination of rows i and j then any of them can be "zeroed" out.

 

[descendency]

R(1) = Row 1. R(i) = Row i. c(j) = the constant that you multiply times row j.

R(j) = c(j-1) * R(j-1) + ... + c(2) * R(2) + c(1) * R(1).

Implies R(j) + (-1)c(j-1) * R(j-1) + ... + (-1)c(2) * R(2) + (-1)c(1) * R(1) = 0.

I'm not sure I am answering your question. That's how we know that the row can be eliminated. We know that the row will be eliminated by the order of which the equations appear. However, it is perfectly legitimate to exchange the orders of the rows and will ultimately change which row is eliminated.