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{\bf Question}

Prove that multiplication of nxn matrices is associative, by
verifying that the ij-th element of $(AB)C$ and $A(BC)$ have the
common value $$\sum_{p,q = 1,\cdots,n} a_{1p} b_{pq} c_{qj}.$$(try
with the ij notation of 2x2 and 3x3 matrices first if you are not
sure what to do for nxn.)

\vspace{.25in}

{\bf Answer}

\begin{eqnarray*} (AB)C_{ij} & = & {\rm ith\ row\ of\ }AB \cdot {\rm
jth\ column\ of\ }C \\ & = & AB_{i1}C_{1j} + AB_{i2}C_{2j}+ \cdots
AB_{in}C_{nj} \\ & = & \sum_{q=1}^n (AB)_{iq} c_{qi} \\ & = &
\sum_{q=1}^n({\rm ith\ row\ of\ }A \cdot {\rm qth column\ of\
}B)c_{qj} \\ & = & \sum_{q=1}^n \left(\sum_{p=1}^n
a_{ip}b_{pq}\right)c_{qj} \\ & = & \sum_{p,q =
1}^na_{ip}b_{pq}c_{qj} \\ {} \\ A(BC)_{ij} & = & {\rm ith\ row\
of\ }A \cdot {\rm jth\ column\ of\ }BC \\  & = & \sum_{p=1}^n
a_{ip} BC_{pj} \\ & = & \sum_{p=1}^na_{ip} \sum_{q=1}^n
b_{pq}c_{qj} \\ & = & \sum_{p,q = 1}^na_{ip}b_{pq}c_{qj}
\end{eqnarray*}


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