Is there a quick/easy way to determine which order you should multiply matrices to combine two transformations? For example you are given two matrices. One which transforms Shape 1 to Shape 2 and the other which transforms Shape 2 to Shape 3. You are asked to find the matrix which transforms Shape 1 to Shape 3. Multiplying both matrices in different orders will give different results, only one result is correct. Rather than checking through your results to find out the correct order, is there a quick/easy way to find out?
If this makes no sense, please tell me what makes no sense. Regards Tom
 A: You look how a single matrix Transformation acts on a vector: If a vector $v$ transforms as $v \rightarrow Av$ under a matrix $A$ then, for some other matrices $A_1,A_2,...$ you have the transformations $v \rightarrow ...A_2 A_1 A v$ (last Transformation first!). If the vector transforms as $v \rightarrow vA$ then by Iteration it follows: $v \rightarrow vAA_1A_2...$ (first Transformation first!).
A: A matrix does not transform some shape into another shape but a full plane (or space) to some other plane (or space). The usual convention is the following (check your manual whether it applies in your case):
Points are coordinate  $n$-tuples $x=(x_1,x_2,\ldots x_n)$. In matrix language these $n$-tuples are written as column vectors
$$[x]:=\left[\matrix{x_1\cr x_2\cr \vdots\cr x_n\cr}\right]\ .$$
A linear map $A:\>{\Bbb R}^n\to{\Bbb R}^m$ throwing points $x\in{\mathbb R}^n$ to image points $y:=Ax\in{\mathbb R}^m$ is encoded in an $(m\times n)$-matrix $[A]$, and the coordinates $(y_1,\ldots, y_m)$ of the image point $y=Ax$, written as a column vector $[y]$, are computed by means of a matrix product:
$$[y]=[A][x]\ .\tag{1}$$
When a second linear map $B:\>{\Bbb R}^m\to{\Bbb R}^d$ is given, throwing points $y\in{\mathbb R}^m$ to image points $z:=By\in{\mathbb R}^d$, then $B$ is encoded in an $(d\times m)$-matrix $[B]$, and in analogy to $(1)$ one has
$$[z]=[B][y]\ .\tag{2}$$
If you want to perform the two maps $$x\mapsto y=Ax \mapsto z=By=B(Ax)$$
in one step then from $(1)$ and $(2)$ and the associativity of maps (resp., matrix multiplication) it follows that
$$[z]=[B]\bigl([A][x]\bigr)=\bigl([B][A]\bigr)[x]=[C][x]$$
with
$$[C]:=[B][A]\ .$$
A: Multiply the matrix which transforms Shape 2 to Shape 3 first. Note: I haven't thoroughly tested it however 99% sure it'll work in all scenarios concerning 2D transformations. I imagine with a 4th shape you would multiply S3-S4 first followed by S2-S3 and S1-S2.
