I realize this is a simple problem, but I'm working through a book and trying to practice the concepts. I have posted a solution in the answers.
Problem: Working over the field with two elements, find the rank of the following matrix: \begin{bmatrix} 1 & 1 & 0 \\[0.3em] 0 & 1 & 1 \\[0.3em] 1 & 0 & 1 \end{bmatrix}