Any injective map from a finite set to itself is surjective. Ax's Theorem extends this to algebraic varieties and regular maps.

Is an injective function also surjective?

If you have an injective function, f(a)≠f(b), so one has to be a and one has to be b, so the function is surjective. The same idea works for sets of any finite size. If the size is n and it is injective, then n distinct elements are in the range, which is all of M, so it is surjective.

Can a matrix be injective but not surjective?

For square matrices, you have both properties at once (or neither). If it has full rank, the matrix is injective and surjective (and thus bijective).