I'm rather new to more formal settings of linear algebra and am having a little bit of trouble acquiring a natural grasp on certain aspects of matrices. I understand that any matrix is congruent (is that the correct term in English?) to a single, unique reduced matrix (RREF). What I'm not too comfortable with is knowing how pivot points are specifically related to individual matrices—one thing I'm struggling with right now:
If I have two matrices with their own defined pivot points and I add the matrices, what effect will this have on the pivot points of the resulting matrix? Is there any relation?
Pivot points can tell you important things about your matrix, for example take a matrix $A_{mxn}$:
If there are $n$ pivot positions (a pivot in every row) in the RREF form, then we know that the columns of the matrix span $\mathbf{R}^{n}$.
If there are $m$ pivot positions (a pivot in every column) in the RREF form then the columns of the matrix are linearly independent.
You can only add two matrices of the same dimensions, say $A_{nxm}$ and $B_{nxm}$. Lets take an example: $ A= \begin{matrix} a & b & c \\ 0 & 0 & f \\ 0 & 0 & 0 \\ \end{matrix} $
$ B= \begin{matrix} a & b & c \\ 0 & e & f \\ 0 & 0 & g \\ \end{matrix} $
You will get the result A+b = $ \begin{matrix} 2a & 2b & 2c \\ 0 & e & 2f \\ 0 & 0 & g \\ \end{matrix} $
The pivot position of each row in the sum is going to be the same position as whichever matrix had a pivot position further left in the row before being added.