Textbook Problem

Let $S$ be the set of four matrices $S=\left\{I,A,B,C\right\},$ where $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ $A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],$ $B=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right],$ $C=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right].$

Follow the procedure described in Exercise 3 of section 1.4 to complete the following multiplication table for $S$. (In this case, the product $BC=A$ is entered as shown in the row with $B$ at the left end and in the column with $C$ at the top.) Is $S$ closed under multiplication in $M_2\left(ℝ\right)?$

$\begin{array}{ccccc}\cdot & I& A& B& C\\ I& & & & \\ A& & & & \\ B& B& C& I& A\\ C& & & & \end{array}$