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      <text>7.2.6.1 The rules of matrix manipulation</text>
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      <text><![CDATA[<h3>The rules of matrix manipulation</h3>
<p>It is worth noting that the process of multiplication can be continued to form products of more than two matrices.</p>
<p>Although two matrices may not commute (i.e. in general \(AB\neq BA\)) the <b>associative law</b> always holds i.e. for matrices which <b>can</b> be multiplied, \[A( BC)=( A B)C.\] The general principle is <b>keep the left to right order</b>, but within that limitation any two adjacent matrices can be multiplied.</p>
<p>It is important to note that it is not always possible to multiply together any two given matrices.</p>
<p>For example if \(A=\begin{bmatrix}1 & 2\\ 3 & 4\end{bmatrix}\) and \(B=\begin{bmatrix}a & b & c\\ d & e & f\end{bmatrix}\) then \[AB=\begin{bmatrix} a+2d & b+2e & c+2f\\ 3a+4d & 3b+4e & 3c+4f\end{bmatrix}\]</p>
<p>However \(BA=\begin{bmatrix}a & b & c\\
d & e & f\end{bmatrix}\begin{bmatrix}1 & 2\\ 3 & 4\end{bmatrix}\) is <b>not defined</b> since each row of \(B\) has three elements whereas each column of \(A\) has two elements and we cannot multiply these elements in the manner described.</p>]]></text>
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