We define the free matrad $\mathcal{H}_{\infty },$ generated by a singleton $\theta _{m}^{n}$ in each bidegree $\left(m,n\right),$ and construct the matrahedra $\left\{KK_{n,m}=KK_{m,n}\right\},$ of which $KK_{n,1}=KK_{1,n}$ is the Stasheff associahedron $K_{n}.$ We realize $\mathcal{H}_{\infty }$ as the cellular chains of $KK$ and define an $A_{\infty }$-bialgebra as an algebra over $\mathcal{H}_{\infty }.$