Mapping class group of a torus

Question

Seems like i have some false understanding that need to be fixed. I have check varies sources saying that the mapping class group of a torus is isomorphic to $GL(2,\mathbb Z)$, however my false intuition tells me it should be $SL(2,\mathbb Z)$. Dehn-Lickorish theorem tells us any orientation preserving homeomorphism of a genus $g$ surface can be generated by a product of Dehn twists along 3g-1 curves, in the case of the torus, the curves are the meridian and the preferred longitude. If we consider a homeomorphisms generated by Dehn twists as induced map of linear transformation in the covering space, then MCG($T^2$) should be isomorphic to the group genenrated by $\left[ \begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array} \right]$, $\left[ \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right]$, where this two matrix corresponds to Dehn twist along meridian and longitude if we compose the matrix with the projection map. But then these two matrices are the generators of $SL(2,\mathbb Z)$, so I thought MCG($T^2$) should be isomorphic to $SL(2,\mathbb Z)$. Could you help me have a look where my understanding went wrong?