my question is about inverse matrix

Question

If $A$ is an $n\times n$ matrix and $A^2=I_n$, find $A^{-1}$.

a) $A^4$
b) $A^2$
c) $A$
d) There is insufficient information.

Answer 1

$$A^2=I_n$$ pre multiplying $A^{-1}$ on both sides, gives $$A=A^{-1}$$

Answer 2

Since $A^2 = I_{n}$, so $AA = I_{n}$, we have $A^{-1} = A$ by definition.