Isometry on the sphere

Question

We know that an isometry $A$ on the sphere is an involution if $A^2=I$. My question would be if the product of two involutions is an involution? I think is not but I do not know how to prove it.

Answer 1

To prove that a statement is false, it suffices to show a counterexample. A matrix represents an involution if and only if it is equal to its inverse (clear!) which is why the matrices

$$M_1=\begin{pmatrix} 7&12\\4&-7\end{pmatrix} \text{ and }\space M_2=\begin{pmatrix} 5&12\\2&-5\end{pmatrix} $$ are involutory (verify this).

But we have $$M_3=M_1*M_2=\begin{pmatrix} 59&24\\6&83\end{pmatrix}$$ If $M_3 $were involutory then it should be equal to its inverse but it is not the case; indeed

$$M_3^{-1}=\frac{1}{4753}\begin{pmatrix} 83&-24\\-6&59\end{pmatrix}\ne M_3 $$