To show how matrix multiplication is implemented, here are two matrixes:
┌ ┐ ┌ ┐
│ a b c │ │ j k l │
│ d e f │ * │ m n o │
│ g h i │ │ p q r │
└ ┘ └ ┘
The multiplication of these two matrixes produces a value for each of the
nine elements of the resulting matrix:
┌ ┐
│ element1 element2 element3 │
│ element4 element5 element6 │
│ element7 element8 element9 │
└ ┘
To produce element 1 of the matrix, element a is multiplied by element j,
element b is multiplied by element m, and element c is multiplied by element
p. That is:
element1 = (a x j) + (b x m) + (c x p)
To produce element 2 of the matrix, element a is multiplied by element k, element b is multiplied by element n, and element c is multiplied by element q. To produce element 3 of the matrix, elements a, b, and c are multiplied by their corresponding elements (l, o, and r) in the third vertical line of the second matrix. To produce element 4 of the matrix, you move down a row in the first matrix. That is, element d is multiplied by element j, element e is multiplied by element m, and element f is multiplied by element p. You continue the multiplication in this way until each of the nine elements has a value. The complete workings for this example are as follows:
element1 = (a x j) + (b x m) + (c x p)
element2 = (a x k) + (b x n) + (c x q)
element3 = (a x l) + (b x o) + (c x r)
element4 = (d x j) + (e x m) + (f x p)
element5 = (d x k) + (e x n) + (f x q)
element6 = (d x l) + (e x o) + (f x r)
element7 = (g x j) + (h x m) + (i x p)
element8 = (g x k) + (h x n) + (i x q)
element9 = (g x l) + (h x o) + (i x r)
Note that if the order of the two matrixes is reversed, the results of the multiplication are different.
Here is a simple example in which an object is scaled by a factor of 3, and then translated by (5,4):
┌ ┐ ┌ ┐ ┌ ┐
│ 3 0 0 │ │ 1 0 0 │ │ 3 0 0 │
│ 0 3 0 │ * │ 0 1 0 │ = │ 0 3 0 │
│ 0 0 1 │ │ 5 4 1 │ │ 5 4 1 │
└ ┘ └ ┘ └ ┘
You can multiply together as many transformation matrixes as you require.