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?