Why can't I prove this set of vectors spans I'm asked if the matrices
$\begin{pmatrix}
1 & 1\\
1 & -1
\end{pmatrix}$,
$\begin{pmatrix}
0 & 2\\
3 & 0
\end{pmatrix}$, $\begin{pmatrix}
2 & 1\\
-3 & -2
\end{pmatrix}$, $\begin{pmatrix}
-1 & 4\\
5 & 1
\end{pmatrix}$
span $M_2 (\mathbb{R})$.
I know that the traces of each of them are $0$ so they can't possibly span $M_2 (\mathbb{R})$ since you can't write them as a linear combination with matrices that have non-zero traces.
However, I also attempted to do this with a system of equations. If I make a coefficient matrix from this, I get:
$\begin{pmatrix}
1 & 0 & 2 & -1 \\
1 & 2 & 1 & 4 \\
1 & 3 & -3 & 5 \\
-1 & 0 & -2 & 1 \\
\end{pmatrix}$.
After row reducing, I get:
$\begin{pmatrix}
1 & 0 & 0 & -13/7 \\
0& 1 & 0& 19/7 \\
0 & 0 & 1 & 3/7 \\
0 & 0 & 0 & 0 \\
\end{pmatrix}$
this means the last variable is free and I have a consistent solution and hence, should span $M_2 (\mathbb{R})$ but it doesn't.
Why is my reasoning wrong?

As the last row is equal to zero, you won't be able to generate a vector having the last coordinate not equal to zero.
Hence those four matrices can't span $M_2(\mathbb R)$.
