What is the intutive explanation of why the notation of matrices is as it is? If I want to solve a system of linear equations, like
2x-y=1
x+2y=4
Then the matrix notation for the same would be:
$$ \begin{bmatrix} 2 & -1 \\ 1 & 2  \\ \end{bmatrix} \begin{bmatrix} X\\ Y\\ \end{bmatrix} =  \begin{bmatrix} 1\\ 2\\ \end{bmatrix}$$
I'd like to know how did this notation come into existence?
Is this notation intutive for everyone? Or is there any significance of this notation? Or was this just proposed by someone (or a set of people) and then set as the standard?

I do not know the origin of the notation but when I see a matrix I recall one lecture from MIT given by G. Strang. He explains what columns and rows of a matrix are. It also shows why the matrix notation is so convenient. The lecture is called The geometry of linear equations and you can watch it here: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-1-the-geometry-of-linear-equations/
