\[R: \mathbf{V}_1 \rightarrow \mathbf{V}_2 , \:S: \mathbf{V}_ \rightarrow \mathbf{V}_3 , \: T: \mathbf{V}_3 \rightarrow \mathbf{V}_4\]
with associated matrices \[R, \: S \: T\]
respectively. Then composition of linear transformations, and matrix multiplication, is associative. That is\[T \circ (S \circ R) \mathbf{v}_1=(T \circ S) \circ R \mathbf{v}_1\]
\[T(SR) \mathbf{v}_1=(TS)R \mathbf{v}_1\]
Considering the left hand side of the first one above
\[S \circ R\]
sends \[\mathbf{V}_1\]
to \[\mathbf{v}_3\]
then \[T\]
sends \[\mathbf{v}_3\]
to \[\mathbf{v}_4\]
.On the right hand side
\[R\]
sends \[\mathbf{v}_1\]
to \[\mathbf{v}_2\]
then \[T \circ S\]
sends \[\mathbf{v}_2\]
to \[\mathbf{v}_4\]
.Hence the first equation holds. Since the matrices represent the corresponding transformations, the second also holds, and in fact,
\[T(SR) \mathbf{v}_1=(TS)R \mathbf{v}_1 =(TSR) \mathbf{v}_1\]