Let M be a set of $(2\times 2)$ non-singular matrices and R be a relation defined on set M such that R={$(\mathrm{A},\mathrm{B});\mathrm{A},\mathrm{B}\in \mathrm{M}$ ; A is inverse of B} then R is

$(\mathrm{A},\mathrm{A})\notin \mathrm{R}$ because it is not necessary that each matrix is inverse of itself.
Let $(\mathrm{A},\mathrm{A})\in \mathrm{R}\Rightarrow$ A & B are inverse of each other.
$$\begin{gathered} \Rightarrow \mathrm{AB}=\mathrm{I}=\mathrm{BA} \\ \Rightarrow \mathrm{BA}=\mathrm{AB}\Rightarrow (\mathrm{B},\mathrm{A})\in \mathrm{R} \\ \mathrm{Now} \mathrm{Let} (\mathrm{A},\mathrm{B})\in \mathrm{R}\Rightarrow \mathrm{AB}=\mathrm{I}=\mathrm{BA} ..\left(1\right) \\ (\mathrm{B},\mathrm{C})\in \mathrm{R}\Rightarrow \mathrm{BC}=\mathrm{I}=\mathrm{CB} ...\left(2\right) \\ \mathrm{From} \left(1\right) \& \left(2\right) \left(\mathrm{AB}\right) \left(\mathrm{BC}\right) = \mathrm{I} \\ {\mathrm{AB}}^2\mathrm{C}=\mathrm{I} \end{gathered}$$
It is clear that ${\mathrm{AB}}^2\mathrm{C}=\mathrm{I}$ is $\mathrm{AC}=\mathrm{I}$ if,  ${\mathrm{B}}^2=\mathrm{I}$
which not possible of each matrix.