TITLE: How to construct a $2\times 2$ real matrix $A$ not equal to Identity such that $A^3=I$?
QUESTION [8 upvotes]: How to construct a $2\times 2$ real matrix $A$ not equal to Identity such
that $A^3=I$?
There is a correspondence between the ring of complex numbers and the
ring  of  $2\times2$ matrices (0 matrix
is included!) i.e.,$$a+ib\leftrightarrow\begin{pmatrix}a&-b\\b&a\end{pmatrix}$$
Can I apply this result and construct such matrix?

REPLY [2 votes]: The matrix $A=\begin{pmatrix} 0&1\\-c&-b\end{pmatrix}$ satisfies $A^2+bA+cI=0$.
So the matrix $\begin{pmatrix} 0&1\\-1&-1\end{pmatrix}$ satisfies $A^3-I=(A-I)(A^2+A+I)=0$