I need to determine whether this matrix is injective \begin{pmatrix} 2 & 0 & 4\\ 0 & 3 & 0\\ 1 & 7 & 2 \end{pmatrix}

Using gaussian elimination, this is what I have done: \begin{pmatrix} 2 & 0 & 4 &|& 0\\ 0 & 3 & 0 &|& 0\\ 1 & 7 & 2 &|& 0 \end{pmatrix}

Divide row1 by 2, and then minus row3 by values of row1: \begin{pmatrix} 1 & 0 & 2 &|& 0\\ 0 & 3 & 0 &|& 0\\ 0 & 7 & 0 &|& 0 \end{pmatrix}

Divide row 2 by 3, divide row 3 by 7 and minus row 3 by row2: \begin{pmatrix} 1 & 0 & 2 &|& 0\\ 0 & 1 & 0 &|& 0\\ 0 & 0 & 0 &|& 0 \end{pmatrix}

Am I doing this correctly? How do I show that the matrix is (not) injective? I was thinking along the lines of "$x + z \ne 0$."