# 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?

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Which notation do you mean exactly? Notation for matrix? or notation for representing system of equations using matrices? – user160738 Jul 6 '14 at 0:54
Perhaps this article on the history of mathematical notation from Wikipedia would be of interest. – Peter Woolfitt Jul 6 '14 at 0:57
@user160738 well, actually both – GrowinMan Jul 6 '14 at 1:03

As in my post here, when matrix theory was developed, this notation was not used. Instead, it looked more like

$$(X,Y,Z)= \left( \begin{array}{ccc} a & b & c \\ a' & b' & c' \\ a'' & b'' & c'' \end{array} \right)(x,y,z)$$

Which represented the set of linear functions $(ax + by + cz, a'z + b'y + c'z, a''z + b''y + c''z)$ which are then called $(X,Y,Z)$.

This is not the exact notation that was used in 1857 (visible at the bottom of the post) but is more historically accurate than the current notation. We would write your problem as

$$(1,2)= \left( \begin{array}{cc} 2 & -1 \\ 1 & 2 \\ \end{array} \right)(X,Y)$$

It is clear what this is stating in the context of matrix equations. $2X-Y = 1$ and $X+2Y =2$. This is very intuitive but did not stand the test of history. As seen by my post, matrix multiplication was discovered and then was denoted by

$$\left( \begin{array}{cc} a & b \\ a' & b' \end{array} \right)\!\!\!\left( \begin{array}{ccc} \alpha & \beta \\ \alpha' & \beta' \end{array} \right) = \left( \begin{array}{cc} a\alpha+b\alpha' & a\beta+b\beta' \\ a'\alpha+b'\alpha' & a'\beta+b'\beta' \end{array} \right)$$

This notation for matrix multiplication is not consistent with the notation for linear systems so that at some point the matrix equations would be written with column vectors (as follows) and would match matrix multiplication.

$$\left( \begin{array}{c} 1 \\ 2 \end{array} \right)= \left( \begin{array}{ccc} 2 & -1 \\ 1 & 2 \end{array} \right)\left( \begin{array}{c} X \\ Y \end{array} \right)$$

In short, this notation of matrices is not the most intuitive but makes the most sense because it matches matrix multiplication in function and form. I can only imagine that having a unified notation reduced confusion while still allowing a semi-intuitive notation.

For reference, this is what the older matrix notation looked like (Source: Memoir on the theory of matrices By Authur Cayley, 1857). If I ever figure out how to typeset this with cross browser compatibility I will edit it in.

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Aaaand, that's it, I'm joining this section of SO just to say that was an amazing and unexpected answer. – amenadiel Jul 6 '14 at 5:08
Very nice answer, but your updated formatting doesn't work for me in Firefox on Mac: i.stack.imgur.com/MBpeB.png ...It's probably best not to use fancy spacing tricks in MathJax because of potential cross-browser rendering inconsistencies. – Rahul Jul 6 '14 at 5:58
@Rahul Thanks for the notification. I will roll it back for the sake of everyone who doesn't use chrome. For reference, it looked like this. – Brad Jul 6 '14 at 6:00

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/

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