It's possible to solve $Ax=b$ recursively where $x, b \in \Re^n$ are vectors and $A \in \Re^{m*n}$ where $m > n$, buy using Recursive Least Squares(RLS).
But what if $AX=B$ where $A \in \Re^{m*n}$ and $X, B \in \Re^{n*k}$ where $k > 1$ and $m > n$. How can I solve that recursively if $A, B$ are known?
Notice that each column of $B$ obtains contributions only from the matching column of $X$, and, conversely, each column of $X$ only contributes to one column of $B$, so you have "number of columns of $B$" simultaneous uncoupled equations in your system.
So the matrix version of your problem appears to be number-of-columns-of-$B$ uncoupled parallel copies of the vector version you describe first.