Say I have the following variable length matrix containing unknown values:
$$ AB=\left (\begin{array}{ccc} ab_{11} & \cdots & ab_{1j}\\ \vdots & \ddots & \vdots \\ ab_{i1} & \cdots & ab_{ij}\\ \end{array}\right) $$
And the following variable length vectors and matrix (according to the size of the preceding matrix) containing known values:
$$ A=\left (\begin{array}{c} a_1 \\ \vdots\\ a_n\\ \end{array}\right) B=\left (\begin{array}{c} b_1 \\ \vdots\\ b_m\\ \end{array}\right) K=\left (\begin{array}{ccc} k_{11} & \cdots & k_{1m} \\ \vdots & \ddots & \vdots \\ k_{n1} & \cdots & k_{nm}\\ \end{array}\right) $$
These are all related by the following equation: $$ AB = [A \cdot B^T - A \cdot 1_n^T \cdot AB - AB \cdot 1_m \cdot B^T + AB \cdot 1_m \cdot1_n^T \cdot AB]\circ K\\ $$
Where $1_i$ is a column vector of length $i$ containing all $1$'s. Given all of the known values, I want to know if it is possible to solve for $AB$. If so how should I go about figuring this out? I only have a basic education on matrix math.