Suppose that $A$ is an $n\times n$ matrix with $n>1$ and $A^2 = I_n$, where $I_n$ denotes the $n\times n$ identity matrix.
a) What are the possible eigenvalues of $A$? (I know that two eigenvalues are $1$ and $-1$, i.e. if $A$ is the positive or negative $n\times n$ identity matrix, but are there others?)
b) Is $A + 15I_2$ invertible? Why or why not? I'm going to say yes, because $-15$ cannot be an eigenvalue so $\det(A+15I_2)$ will not be $0$? Not sure.
c) Give an example of a $2\times 2$ matrix $A$ with $A^2 = I_2$, but $A$ does not equal $I_n$ and A does not equal $-I_2$. I found one, $$A=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$
Suppose $A^2=I_n$ and $v\neq 0$ is an eigenvector of $A$ with eigenvalue $\lambda$ then we have: $$Av=\lambda v\implies A^2v=A\lambda v\implies I_nv=\lambda(Av)\implies v=\lambda^2v\implies v(\lambda^2-1)=0$$ since $v\neq 0$, we have $\lambda =\pm 1$, which is what you have.
For part (b), that is a good answer on your part, and the fact that $-15$ is not an eigenvalue of $A$ follows from the above proof.
For part (c), that is a good example.