So I'm going over some exercises in my linear algebra textbook and I am confused with question 7c.
I've looked over a few solutions online, and all of them conclude that all entries are zero and don't expand much further on why except with using the fact that row 3 = row 1 + row 2. I feel like this isn't completely intuitive to me and maybe I am misunderstanding the definition of "elimination". When I think of elimination in this case, I am thinking of eliminating it via the process of row operations to achieve a diagonal (or upper triangle) matrix to receive a solution. So in this case, I am envisioning that row 3 will look like (0 0 1). However, all row 3 entries become zero when eliminating equation 3.
Best Answer
You are right that elimination is via the process of row operations, such as subtracting one row from another. If row 3 = row 1 + row 2, and then we subtract row 1 and row 2 from row 3, what is left?
A simple example illustrating what happens is the following matrix:
\begin{bmatrix}1&0&1\\0&1&1\\1&1&2\end{bmatrix}