Chapter 4.5, Problem 1ED

The only real number solutions to the equation $x^2=1$ are $x=1\text{ and }x=-1.$

(A) Show that $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ satisfies $A^2=I$ where $I$ is the $2\times 2$ identity.

(B) Show that $B=\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right]$ satisfies $B^2=I.$

(C) Find a $2\times 2$ matrix with all elements nonzero whose square is the $2\times 2$ identity matrix.