<s>[INST] How do I show that a matrix is injective? 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 "$x + z \ne 0$."
 [/INST] Look at vectors of the form $\left(\begin{array}{c}2x\\ 0\\ -x\end{array}\right)$.  What does your matrix do to these vectors?  What does injective mean?
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