Why is this equation converted to matrix form in this way? Is it possible to multiply an inverse matrix with a vector?

I have been banging my head on wall for days trying to decode this equation. please help me out with this…

Below is the equation (consider $x$ as $Delta x$, and $y$ as $Delta y$):

$x = – eta(Id-eta^2D^2_{xy}fD^2_{yx}g)^{-1}(nabla_xf-eta D^2_{xy}fnabla_yg)$

$y = – eta(Id-eta^2D^2_{yx}gD^2_{xy}f)^{-1}(nabla_yg-eta D^2_{yx}gnabla_xf)$

The authors of the paper have converted it into matrix form as:

I am failing to see how is this possible, I mean how am I supposed to multiply inverse of a matrix with a vector.

Thanks in advance for helping me out, and forgive my poor english. I have been struggling in this equation for days, it is from the paper competitive gradient descent (for someone who might be interested equation number 3 and 4) https://arxiv.org/abs/1905.12103?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%253A+arxiv%252FQSXk+%2528ExcitingAds%2521+cs+updates+on+arXiv.org%2529.