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Matrix multiplication is not straightforward and an important thing to remember is that it is noncommutative (i.e. $AB{\ne}BA$ )
To multiply two matrices together you sort of multiply rows by columns. However, in order to multiply two matrices together, the matrix being postmultiplied must have the same number of columns as the matrix being premultiplied has rows (e.g. you cannot postmultiply a 3x2 matrix by a 3x4 matrix, but you can postmultiply a 2x3 matrix by a 3x4 matrix: the result will be a 2x4 matrix).
In general...
$\left[\begin{matrix}a_1&a_2&a_3\\b_1&b_2&b_3\end{matrix}\right]\left[\begin{matrix}c_1&c_2\\d_1&d_2\\e_1&e_2\end{matrix}\right]=\left[\begin{matrix}a_1c_1+a_2d_1+a_3e_1&a_1c_2+a_2d_2+a_3e_2\\b_1c_1+b_2d_1+b_3e_1&b_1c_2+b_2d_2+b_3e_2\end{matrix}\right]$
Another way to look at this.

So $(a_1,a_2).(x,y)~=~x'.$
Similarly, $(b_1,b_2).(x,y)~=~y'.$
As a notational convenience we write these one above the other, like this:
We've put square brackets to help show how things are organised, and that has allowed us to write the $(x,y)$ only once, thereby avoiding repetition.
There's an inconsistency here, though, because the vector $(x,y)$ is written horizontally, and the vector $(x',y')$ is written vertically, even though each represents a point in the plane. So we rotate the $(x,y)$ around and get this:
This is just a way of writing down the original equations, arranged to be compact. The 2x2 matrix holds the coefficients of the transformation.
Now there are two ways to extend this.


This all needs to be reworked.
Mathematicians are lazy, and they want, indeed need, simple, elegant ways to write things down and work with them. Instead of working with the equation forms of these transformations it's much easier to work with the matrix forms, provided we use definitions like this that may at first appear arbitrary and unmotivated, but turn out to be concise encapsulations of otherwise nasty calculations.
DeterminantOfAMatrix DifferenceOfTwoSquares MathematicsTaxonomy 
(none)  LinearAlgebra  
Matrices  EquationOfALine LinearFunction MatrixTransformation 

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MatrixTransformation  (none)  Adding2DVectors AddingVectors CategoryMetaTopic CommutativeOperation MagnitudeOfAVector UnitVector 
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