Let $B$ be a bialgebra, possibly noncommutative, over a field $k$ and $G = (g^i_j)^{i = 1,\ldots, n}_{j = 1,\ldots, n}$ an $n\times n$-matrix over $B$. $B$ is a matrix bialgebra with basis $G$ if
the set of entries of $G$ generates $B$ and
the comultiplication $\Delta$ and counit $\epsilon$ satisfy the matrix equations $\Delta G = G \otimes G$ (i.e. in components $\Delta g^i_j = \sum_{k = 1}^n g^i_k \otimes g^k_j$) and $\epsilon G = 1$ (reading in components $\epsilon (g^i_j) = \delta^i_j$).
According to a result of Redford every finite-dimensional Hopf algebra over a field is a matrix Hopf algebra with respect to some basis.
The free (noncommutative) associative algebra $F$ on $n^2$ generators $f^i_j$ has a unique coalgebra structure making it a matrix bialgebra with basis $(f^i_j)^{i = 1,\ldots, n}_{j = 1,\ldots, n}$. We call it the free matrix bialgebra of rank $n^2$. Every bialgebra quotient of that bialgebra is a matrix bialgebra.
A matrix Hopf algebra $\mathcal{H}$ with basis $T = (t^i_j)$ is a Hopf algebra which possess a matrix subbialgebra $B$ with basis $T$ such that the map $H(id_B):H(B)\rightarrow\mathcal{H}$ is onto (where $H(B)$ denotes the Hopf envelope of $B$ and $H$ is understood as a functor).
A matrix Hopf algebra $\mathcal{H}$ with basis $T$ is often not a matrix bialgebra with basis $T$: e.g. the commutative coordinate ring of $GL(n,k)$ is not a matrix bialgebra with respect to the obvious basis $T$; in this example this can be repaired by enlarging the basis by one group-like element: the inverse of the determinant. On the other hand, the coordinate algebra of the special linear group $O(SL(n,k))$ is a matrix bialgebra and a matrix Hopf algebra with the same standard basis $T$.