Showing that a C-infinity function exists

I am running into some trouble proving this statement:

There is a $\displaystyle C^{\infty}$ square-root function in a neighborhood of the identity matrix $\displaystyle I$ in $\displaystyle Mat(n\times n,R)$.

Basically I need to show that there is a $\displaystyle C^{\infty}$ function $\displaystyle f$ from a neighborhood $\displaystyle U$ of $\displaystyle I$ into $\displaystyle Mat(n\times n,R)$, with $\displaystyle f(I)= I$, such that $\displaystyle f(A)^{2}= A$ for all $\displaystyle A\in U$.

I know that I can apply the inverse-function theorem with $\displaystyle F(A)=A^{2}$. But I am unsure where to begin.