Question

Is matrix multiplication associative?

Answer 1

Determine if matrix multiplication is associative

Let us consider three matrices $A,B,C$ and check the associative property of matrix.

Let, $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, $B=\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]$ , $C=\left[\begin{array}{cc}i& j\\ k& l\end{array}\right]$

The associative property states that: $(A.B).C=A.(B.C)$

Checking for $LHS$:

$$\begin{gathered} \Rightarrow (A.B).C \\ \Rightarrow \left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\times \left[\begin{array}{cc}e& f\\ g& h\end{array}\right]\right)\times \left[\begin{array}{cc}i& j\\ k& l\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}ae+bg& af+bh\\ ce+dg& cf+dh\end{array}\right]\left[\begin{array}{cc}i& j\\ k& l\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}(ae+bg)i+(af+bh)k& (ae+bg)j+\left(af+bh\right)l\\ \left(ce+dg\right)i+\left(cf+dh\right)k& \left(ce+dg\right)j+\left(cf+dh\right)l\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}eia+fja+gib+hkb& eja+fla+gjb+hlb\\ eic+fjc+gid+hkd& ejc+flc+gjd+hld\end{array}\right] \end{gathered}$$

Checking for $RHS$:

$$\begin{gathered} \Rightarrow A.(B.C) \\ \Rightarrow \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\times \left(\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]\times \left[\begin{array}{cc}i& j\\ k& l\end{array}\right]\right) \\ \Rightarrow \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\times \left[\begin{array}{cc}ei+fj& ej+fl\\ gi+hk& gj+hl\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}(ei+fj)a+(gi+hk)b& (ej+fl)a+\left(gj+hl\right)b\\ \left(ei+fj\right)c+\left(gi+hk\right)d& \left(ej+fl\right)c+\left(gj+hl\right)d\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}eia+fja+gib+hkb& eja+fla+gjb+hlb\\ eic+fjc+gid+hkd& ejc+flc+gjd+hld\end{array}\right] \end{gathered}$$

Hence, $\text{LHS = RHS}$ has been proved.

Therefore, we can conclude that matrix multiplication is associative.