Question

Show that the set of four matrices $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$ form an abelian group, under multiplication of matrices.

Answer 1

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I$ , $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]=-I$ . Let $\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]=a$ and $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]=b$

Given matrices are invertible, Therefore they satisfy invertibility property

Observe that given matrices satisfies commutative property $($ $Ia=aI=a$ , $bI=Ib=b$ , $-Ia=a(-I)=-a$ , $-Ib=b(-I)=b$ , $ab=ba$ , $I(-I)=(-I)I=-I$ $)$

Therefore it also satisfies associative property

Hence the given matrices form an abelian group , under multiplication of matrices.