1. Let R1 and R2 be in RREF and be equivalent to the same matrix A after row operations.
Thus, R1 * x = 0 and R2 * x = 0 have the same solutions. (Because solutions stay the
same if the two matrices can reach each other through row operations). We know that in
RREF, every pivot column has a leading 1. If R1 and R2 had different pivot columns,
then different variables would be free which would lead to different solution sets. Since
we know that they are the same, then we know the pivot columns must be the same.
Once we know that pivot columns must be the same, we know that all of the pivot
columns must be identical because all other entries in that column are 0. The non-pivot
columns tell you how to express the pivot variables in terms of the free ones, so we also
know that the non pivot columns must also be identical because they must give the
same values to complete the solution set. Therefore the two matrices R1 and R2 are
equal and RREF is unique.
2. We know that A can be transformed into R through row operations. We also know that B
can be transformed into R through row operations. Because row operations are
reversible, we know that R can be transformed into B through row operations. So
therefore, A can be transformed into R which can be transformed into B, and therefore A
can be transformed into B through row operations.