TITLE: Combine transformation matrices
QUESTION [6 upvotes]: Question:
Find the transformation matrix that combines the following transformation matrices, in order:
$$\begin{bmatrix}
 &3  &0  &0  &0 \\ 
 &0  &-1  &0  &0 \\ 
 &0  &0  &2  &0 \\ 
 &0  &0  &0  &1 \\  
\end{bmatrix}$$
$$\begin{bmatrix}
 &\frac{\sqrt3}{2}  &0  &\frac{-1}{2}  &0 \\ 
 &0  &1  &0  &0 \\ 
 &\frac{1}{2}  &0  &\frac{\sqrt3}{2}  &0 \\ 
 &0  &0  &0  &1 \\  
\end{bmatrix}$$
$$\begin{bmatrix}
 &1  &0  &0  &3 \\ 
 &0  &1  &0  &-1 \\ 
 &0  &0  &1  &2 \\ 
 &0  &0  &0  &1 \\  
\end{bmatrix}$$
I know you are meant to multiply them together, and I get the same result online calculators get, but the answer in my textbook is:
$$\begin{bmatrix}
 &\frac{3\sqrt3}{2}  &0  &-1  &3 \\ 
 &0  &-1  &0  &-1 \\ 
 &\frac{3}{2}  &0  &\sqrt3  &2 \\ 
 &0  &0  &0  &1 \\  
\end{bmatrix}$$
Am I missing something, how is this result achieved?

REPLY [7 votes]: Hint: order matters in matrix multiplication, and particularly, consider the order in which the transformations are applied. If the matrices are named $A, B, C$ respectively, and operate on some matrix $Q$, which transformation matrix do you want apply first? Second? Lastly?