History of the inverse matrix

I know the definition and the procedure to calculate the inverse of a matrix, but I want to know the history of starting the idea of an inverse matrix. mathematicians must have faced a real life situation which compelled them to invent the inverse matrix. What was the situation?

• I am inclined to believe that mathematicians must have invented matrices because of a need for them in real life. Like so many branches of maths,linear algebra is inherently beautiful. It is high time that the history of maths be pursued seriously by our younger generation. Hopefully more maths is invented for the advancement of mankind. – user3278 Nov 22 '15 at 11:20

Cayley defines the inverse $M^{-1}$ of a matrix in the very same note Remarques sur la notation des fonctions algébriques (1855) where he first introduces matrix multiplication. He emphasizes the fact that a linear system $Mx = \xi$ is then solved by $x = M^{-1}\xi$, and that seems motivation enough for him. Here, $x,\xi\in \mathbb{R}^n$ and $M\in GL(n,\mathbb{R})$.
• Cayley was aware of Cramer's formulas for solving linear systems, and he gave the corresponding formulas for the entries of $M^{-1}$. But as far as motivation, yes, it seems to have been the analogy with solving linear equations in one variable by multiplying by the reciprocal. – Conifold Feb 5 '15 at 18:37