Associativity of Vector and Matrix Background


*

*$A$ is a matrix

*$B$ is a matrix

*$\bf{x}$ is a vector


Question:
Does $(AB)\bf{x}$ $=$ $A(B \bf{x})$ ?
 A: The equality does hold.
$$\left(\left(\begin{matrix}
a_{11} & \cdots & a_{1n}\\
\vdots & \ddots & \vdots\\
a_{n1} & \cdots & a_{nn}\end{matrix}\right)
\left(\begin{matrix}
b_{11} & \cdots & b_{1n}\\
\vdots & \ddots & \vdots\\
b_{n1} & \cdots & b_{nn}\end{matrix}\right)
\right)
\left(\begin{matrix}
x_1\\
\vdots\\
x_n\end{matrix}\right)
$$
$$=\left(\begin{matrix}
\sum_{i=1}^n a_{1i}b_{i1} & \cdots & \sum_{i=1}^n a_{1i}b_{in}\\
\vdots & \ddots & \vdots\\
\sum_{i=1}^n a_{ni}b_{i1} & \cdots & \sum_{i=1}^n a_{ni}b_{in}\end{matrix}\right)
\left(\begin{matrix}
x_1\\
\vdots\\
x_n\end{matrix}\right)
$$
$$
=\left(\begin{matrix}
\sum_{j=1}^n x_j\sum_{i=1}^n a_{1i}b_{ij}\\
\vdots\\
\sum_{j=1}^n x_j\sum_{i=1}^n a_{ni}b_{ij}\\\end{matrix}\right)
$$
$$
=\left(\begin{matrix}
\sum_{j=1}^n \sum_{i=1}^n x_ja_{1i}b_{ij}\\
\vdots\\
\sum_{j=1}^n \sum_{i=1}^n x_ja_{ni}b_{ij}\\\end{matrix}\right)
$$
$$\left(\begin{matrix}
a_{11} & \cdots & a_{1n}\\
\vdots & \ddots & \vdots\\
a_{n1} & \cdots & a_{nn}\end{matrix}\right)
\left(\left(\begin{matrix}
b_{11} & \cdots & b_{1n}\\
\vdots & \ddots & \vdots\\
b_{n1} & \cdots & b_{nn}\end{matrix}\right)
\left(\begin{matrix}
x_1\\
\vdots\\
x_n\end{matrix}\right)\right)
$$
$$=\left(\begin{matrix}
a_{11} & \cdots & a_{1n}\\
\vdots & \ddots & \vdots\\
a_{n1} & \cdots & a_{nn}\end{matrix}\right)
\left(\begin{matrix}
\sum_{i=1}^nb_{1i}x_i\\
\vdots\\
\sum_{i=1}^nb_{ni}x_i\end{matrix}\right)
$$
$$=\left(\begin{matrix}
\sum_{j=1}^na_{1j}\sum_{i=1}^nb_{ji}x_i\\
\vdots\\
\sum_{j=1}^na_{nj}\sum_{i=1}^nb_{ji}x_i\end{matrix}\right)
$$
$$=\left(\begin{matrix}
\sum_{j=1}^n\sum_{i=1}^na_{1j}b_{ji}x_i\\
\vdots\\
\sum_{j=1}^n\sum_{i=1}^na_{nj}b_{ji}x_i\end{matrix}\right)
$$
The variables $i$ and $j$ in the sum are dummy variables. Therefore, we can interchange them to see that the two results are the same.
