Say you have a rotation matrix $R$ and a translation matrix $t$, you can trivially have a single matrix $[\;R\;|\;t\;]$. Now say you have another matrix $R'$, can you write $R'[\;R\;|\;t\;]$ as $[\;R''\;|\;t'\;]$? I would think yes, but I would like to be sure.
Edit: $[\;R\;|\;t\;]$ is a rotation matrix with the right most colomn a translation. It's the same as applying the rotation $R$ first followed by the translation $t$. This might indeed be somewhat of a funky notation but it is used in the following book: http://szeliski.org/Book/. For example on page 50.
Yes, you can.I don't know.Originally, I assumed you meant applying the operations as rotation-rotation-translation, but since you mean rotation-translation-rotation, my reasoning does not apply to your problem. I'll leave it up here anyway, for history.
Think of it in terms of general matrix multiplication: the order of the multiplications on the left hand side here is unimportant, since matrix multiplication is associative.
In other words, if you are transforming a vector $v$, you have the first transformation as $v'=Rtv$. Applying the second rotation, $R'$, you obtain $R'v' = R'(Rtv) = (R'R)tv = R''tv$, again because of the associativity of matrix multiplication.
Note, though, that the order in which you apply these transformations is important - matrix multiplication is in general not commutative, and rotational matrices are no exception.