i was given three vectors and I put them into matrix to check if they are independent or dependent $$\begin{pmatrix} 1&1&3\\ 1&-1&1\\ 1&2&4\\ \end{pmatrix}$$ i used row of reduction and i have $$\begin{pmatrix} 1&1&3\\ 1&-1&1\\ 0&3&3\\ \end{pmatrix}$$ row 3 divide by 3 then add row 2 $$\begin{pmatrix} 1&1&3\\ 0&-1&1\\ 0&0&2\\ \end{pmatrix}$$row 3 divide by 2 then row 1 add row 2 $$\begin{pmatrix} 1&0&4\\ 0&-1&1\\ 0&0&1\\ \end{pmatrix}$$ further reduction i get this eventually $$\begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}\begin{pmatrix}0\\0\\0\end{pmatrix}$$
my conclusion is it is independent but the answer from my lecturer is they are linearly dependent because $2x_1+x_2-x_3=0$
I was confused now. what is exactly mean by independent and dependent
Your 3. matrix is not correct.