Transformation matrix for two rotations If I am required to compute the full transformation matrix compromising of the following sequence of operations:


*

*rotation by $30$ degrees about $x$-axis

*translation by $1$, $-1$, $4$ in $x$, $y$ and $z$, respectively

*rotation by $45$ degrees about $y$ axis


Can I compute the rotation, translation and rotation matrix or would I be required to compute the rotation, rotation and translation matrix?
 A: In the comments you confirmed that your question is whether or not the order of transformations in the sequence has to be kept.
The answer is "yes, it does."  Rotations do not always commute with translations, so it would be disastrous to scramble their order.
Since this question has been lingering for a long time, I decided to create an answer. I had to make some assumptions since you were not completely specific as to the directions of your rotations. In your case we have (using affine $4\times 4$ matrices acting on the left of column vectors)


*

*$\begin{bmatrix}
1 &0&0&0\\
0&\sqrt{3}/2&-1/2 &0\\
0&1/2 &\sqrt{3}/2 &0\\
0&0&0&1
\end{bmatrix}$  (counterclockwise 30 degrees around x)

*$\begin{bmatrix}
1 &0&0&1\\
0&1&0 &-1\\
0&0 &1 &4\\
0&0&0&1
\end{bmatrix}$ (add $(1,-1,4)$ to the x,y,z components)

*$\begin{bmatrix}
\sqrt{2}/2&0&-\sqrt{2}/2 &0\\
0 &1&0&0\\
\sqrt{2}/2&0 &\sqrt{2}/2 &0\\
0&0&0&1
\end{bmatrix}$ (counterclockwise rotation around $y$)


Multiplying them in the order they are applied, (if the matrices above are $A,B,C$, then that order is $CBA$ in my notation) that makes a final transformation of 
$\begin{bmatrix}
\sqrt{2}/2&-\sqrt{2}/4&-\sqrt{6}/4&-3\sqrt{2}/2\\
0&\sqrt{3}/2&-1/2&-1\\
\sqrt{2}/2&\sqrt{2}/4&\sqrt{6}/4&5\sqrt{2}/2\\
0&0&0&1
\end{bmatrix}$
With the same matrices, one can see that changing the order of multiplication changes this final transformation.
