Finding a single vector that is a combination of two $3\times3$ transformations

Question

The transformations $T_1$ and $T_2$ are defined by the matrices $\begin{pmatrix}4&1&1 \\ 1&2&-1\\3&1&1\end{pmatrix}$ and $\begin{pmatrix}1&1&1 \\ 1&2&-1\\0&1&2\end{pmatrix}$ respectively. If $T_1$ transforms a body $S$, volume $10$ cubic units, to its image $S'$ and $T_2$ transforms $S'$ to $S''$, find the single matrix that will transform $S$ to $S''$.

Because $T_2$ will act on the result of the $T_1$ transformation, I thought that the single vector would be the product of both:

$$ T_1 \cdot T_2 = \begin{pmatrix}5&7&5 \\ 3&4&-3\\4&6&4\end{pmatrix} $$

But the correct answer is $\begin{pmatrix}8&4&1 \\ 3&4&-2\\7&4&1\end{pmatrix}$. Is my approach incorrect or my method?

Answer 1

We know that $T_1(S) = S'$ and $T_2(S') = S''$. Substituting, we see that $T_2(T_1(S)) = S''$. But by the associativity of matrix multiplication, we know that $(T_2T_1)(S) = S''$.

Answer 2

$T_1$ acts first, then $T_2$. That is, if you starting with an arbitrary column vector $v$, you would first calculate $T_1v$ then have $T_2$ act on that, hence ... see it now?