Consider a set $S = \{A \in \mathcal{M}_{2x2} \colon A^2 = -I_2\}$, where A is a matrix of complex numbers and $I_2$ is the identity matrix. What are all the elements of $D= \{\det(A) \colon A \in S\}$?
So, this question is saying the set is composed by all the complex matrices such that, when squared, are the symmetric of the identity matrix. And we are supposed to find the determinants where each of the matrices are in $S$. I'm a bit lost. How do I proceed?
Hint If $A^2=-I_2$ then $$ \det(A^2)=\det(-I_2) $$