  In this paper we study the geometry of the set of real square roots of $\pm
I_2$. After some introductory remarks, we begin our study by deriving by quite
elementary methods the forms of the real square roots of $\pm I_2$. We then
discuss the interpretations of these square roots as transformations of the
cartesian $(x,y)$-plane. To study the geometry of the set of square roots of
$\pm I_2$ we consider a slightly more general set of square matrices of order
$2$ and show that these sets are hyperboloids of one sheet or hyperboloids of
two sheets. From these general results we conclude that the set of involutory
matrices of order 2 is a hyperboloid of one sheet and the set of
skew-involutory matrices of order 2 is a hyperboloid of two sheets. The
relations between the geometrical properties of the hyperboloids and the set of
square roots of $I_2$ are also investigated. We then proceed to obtain the
forms of the involutory matrices of order 2 by more advanced methods. We have
considered two approaches: in the first approach we use the concept of a
function of a matrix and in the second approach we use concepts of
split-quaternions.
