Can someone please help me calculate the reduced row echelon form of the following matrix: $$ \begin{bmatrix} 1&1&1&0 \\ 1&1&0&1 \\ 0&0&1&1 \end{bmatrix} \in M_{3,4}(F_2)$$
Where $F_2$ denotes the field of scalars $\{0,1\}$ with operations doen using mod $2$ arithmetic.
I am having problems because no matter what I do, I get no leading entry in column 2. For instance, adding $R_1$ to $R_2$ would make $R_2= \{0,0,1,1\}$.
Can there be no leading entry in the second column of second row? From what I have learned, each column must have a leading entry except for in the bottom row.
Best Answer
After adding the first row to the second one, you get, as you wrote:$$\begin{bmatrix}1&1&1&0\\0&0&1&1\\0&0&1&1\end{bmatrix}.$$Then, after adding the second row to the first and to the third ones, you get:$$\begin{bmatrix}1&1&0&1\\0&0&1&1\\0&0&0&0\end{bmatrix}.$$And this matrix is in RREF.