Define $\mathbf{f}(A) = A^2$, for $A \in \mathbb{R}^{n \times n}$.
(a) Applying the Inverse Function Theorem, show that every matrix $B$ in a neighbourhood of $I$ has (atleast) 2 square roots $A$, that is $A^2 = B$, each varying as a $C^1$ function of $B$.
(b) Can you decide if there are precisely 2 or more ? (Hint: in 2x2 case, what is $D\mathbf{f}\left ( \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \right )$ ?
Now I can apply the inverse function theorem to $\mathbf{f}(A)$ at $A =I$ to conclude the inverse exists. But I cannot see how to apply it and ``show that every matrix $B$ in a neighbourhood of $I$ has (atleast) 2 square roots $A$" is to be solved and the bit about number of square roots.
I would appreciate some hints. This is a problem from Ted Shifrin's book on Mutivariable Mathematics.