<s>[INST] How to construct a $2\times 2$ real matrix $A$ not equal to Identity such that $A^3=I$? 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?
 [/INST] Find the three cube roots of $1$.  Let $a+bi$ be one of those.  They are solutions of $x^3=1$.  Hence $x^3-1=0$.
Since $1$ is one of the solutions, $x-1$ must be one of the factors, thus:
$$
x^3-1 = (x-1)(\cdots\cdots\cdots).
$$
Fill in the blanks by doing long division.  You should get
$$
(x-1)(x^2+x+1).
$$
So the equation is
$$
(x-1)(x^2+x+1) = 0.
$$
That implies
$$
x-1=0\quad\text{or}\quad x^2+x+1 = 0.
$$
Solve the quadratic equation.
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