Matrix Inversion in the complexity class $P$ seen as a decision problem If the set $P$ is defined as the set of decision problems that can be solved by a deterministic Turing Machine in polynomial time, and matrix inversion using Gaussian elimination is $O (n^3)$, then how can I relate these two concepts to conclude that inverting a matrix is in $P$? 

I suppose I need a way of converting a description of Gaussian elimination into a decision problem? Or maybe I am confused about the fundamentals
Gaussian elimination is an algorithm, you can’t frame it as a problem and thus it can’t be in $P$.

But matrix inversion itself is a problem and its decision version is as follows: “does matrix $M$ have an inverse?”

Gaussian elimination then is one possible way to prove that you can always answer that question in polynomial time, therefore matrix inversion is in $P$.