Chapter 1.4, Problem 6E

Prove the associative law of multiplication for $2\times 2$ matrices; that is, let

$\begin{array}{ccc}\begin{array}{cc}A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right],& B=\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right],\end{array}& C=\left[\begin{array}{cc}{c}_{11}& c_{12}\\ c_{21}& c_{22}\end{array}\right]& \end{array}$

and show that

$(AB)C=A(BC)$