Define $f(A) = A^2$ where A is a $n\times n$ matrix.
(a) Show that every matrix $B$ in a neighbourhood of the identity matrix $I$ has at least 2 square roots, that is, each varying as a $C^1$ function of $B$.
(b) Are there exactly 2 or more? (Hint: think about $Df\left ( \begin{bmatrix} 1 & 0 \\
0 & -1 \end{bmatrix} \right )$
I know this problem has already been answered here but I can't understand the answer that was given there.
I know that we have to apply the inverse function theorem, and for the inverse function theorem to be applicable, the determinant of $Df(I)$ has to be zero. But I don't know what does $Df(I)$ look like and now I'm stuck.