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Associate Property:
Let

(1)
\begin{pmatrix} a & b \\c & d\end{pmatrix}, \begin{pmatrix}e & f \\g & h\end{pmatrix}, and \begin{pmatrix}i & j \\k & l \end{pmatrix}

be elements of GL2(R)
Since GL2(R)is closed under mult, we know:

(2)
\begin{align} (\begin{pmatrix}a & b \\c & d\end{pmatrix} \begin{pmatrix}e & f \\g & h\end{pmatrix}) \begin{pmatrix}i & j \\k & l\end{pmatrix} = \begin{pmatrix}ae+bg & af+bh \\ce+dg & cf+dh \end{pmatrix} \begin{pmatrix}i & j \\k & l\end{pmatrix} = \begin{pmatrix}iae+ibg+kaf+kbh & jae+jbg+laf+lbh \\ice+idg+kcf+kdh & jce+jdg+lcf+ldh\end{pmatrix} \end{align}

and we know:

(3)
\begin{pmatrix} a & b \\c & d\end{pmatrix} (\begin{pmatrix}e & f \\g & h\end{pmatrix} \begin{pmatrix}i & j \\k & l\end{pmatrix}) = \begin{pmatrix}a & b \\c & d\end{pmatrix} \begin{pmatrix}ei+fk & ej+fl \\gi+hk & gj+hl\end{pmatrix} = \begin{pmatrix}aei+afk+bgi+bhk & aej+afl+bgj+bhl \\cei+cfk+dgi+dhk & cej+cfl+dgj+dhl \end{pmatrix}

Since R is communitive under addition and multiplication

(4)
\begin{pmatrix} iae+ibg+kaf+kbh & jae+jbg+laf+lbh \\ice+idg+kcf+kdh & jce+jdg+lcf+ldh\end{pmatrix}= \begin{pmatrix}aei+afk+bgi+bhk & aej+afl+bgj+bhl \\cei+cfk+dgi+dhk & cej+cfl+dgj+dhl \end{pmatrix}

by substitution

(5)
\begin{align} (\begin{pmatrix}a & b \\c & d\end{pmatrix} \begin{pmatrix}e & f \\g & h\end{pmatrix}) \begin{pmatrix}i & j \\k & l\end{pmatrix} = \begin{pmatrix}a & b \\c & d\end{pmatrix} (\begin{pmatrix}e & f \\g & h\end{pmatrix} \begin{pmatrix}i & j \\k & l\end{pmatrix}) \end{align}

Therefore, GL2(R) is Associative

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