Need help to prove that a given matrix has a given rank.

Question

For any positive integer $k$, there is a matrix $M_k$ of size $2^k\times 2^k$ such that each rows and columns of $M_k$ correspond to some subset of $2^{[k]}$. Let $M_k[S][S']$ $(S,S'\subseteq [k])$ denote the element in the row corresponding to subset $S$ and column corresponding to subset $S'$ of $M_k$. $M_k$ is defined such that $M_k[S][S']=1 $ if $2|S\cap S'|=|S'|$ otherwise zero. I have to prove that the rank of $M_k$ is $2^{k-1}$.
It can be seen that columns corresponding to subsets of odd size have only zeros and rows corresponding to any subset $S$ and $[k]\setminus S$ are equal. So, now a submatrix of $M$ of size $2^{k-1}\times 2^{k-1}$ can be formed by taking columns corresponding to subsets of even size and by taking only one of the two rows corresponding to any subset $S$ or $[k]\setminus S$. Now, the problem is reduced to proving that this new matrix is full rank.