How can one prove that the number of images formed by the reflecting surfaces of two plane mirrors at right angles to each other is 3? How can one prove that the number of images formed by two plane mirrors at right angles to each other is 3?
Is there a mathematical proof for the same? 
 A: You can just draw what the rays look like for a particular observer. That's a fine proof.
A: Sticking to 2D for simplicity, the transformation matrices for reflections in the x = 0 and y = 0 lines are:
$$ R_x = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right) $$
$$ R_y = \left( \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right) $$
Any combination of these transformations can be given by $R_x^m R_y^n$ where $m$ and $n$ are integers giving the number of each reflection.
But both the reflections are their own inverses i.e. $R_xR_x = I$ and $R_yR_y = I$. If this isn't intuitively obvious you can prove it by multiplying out the matrices above. So for any integer $m$, $R^m$ is equal to $R$ if $m$ is odd or $I$ if $m$ is even. That means there are only three distinct combinations that are not the identity:


*

*$R_x$

*$R_y$

*$R_xR_y$


That's why there are three and only three reflections.
A: It is very simple to prove it mathematically.
You can use the formula:
$$n = \frac{360⁰}{θ} -1$$
