  We complete the solution of the problem of finding the universal central
extension of the matrix superalgebras $\mathfrak{sl}(m, n, A)$ where $A$ is an
associative superalgebra and computing $H_2\big(\mathfrak{sl}(m, n, A)\big)$.
The Steinberg Lie superalgebra $\mathfrak{st}(m, n, A)$ has a very important
role and we will also find out $H_2\big(\mathfrak{st}(m, n, A)\big)$. In Chen
and Sun (arXiv:1311.7079, 2013) it is solved the problem where $m+n \geq 5$ and
in Chen and Guay (Algebr. Represent. Theory, 2013) it is solved when $n=0$, so
here we work out the three remaining cases $\mathfrak{sl}(2, 1, A),
\mathfrak{sl}(3, 1, A)$ and $\mathfrak{sl}(2, 2, A)$.
