Local inversion theorem problem Let $f:$ $M_2(\mathbb{R})\rightarrow M_2(\mathbb{R})$ defined by $f(A)=A^2$. In the neighbourhoods of what diagonal matrix is f a local diffeomorphism? 
 A: You know that if $f$ is smooth, then for $f$ to be a local diffeomorphism around a point $p$ it is sufficient that $\mathrm{Jac}(f)$ is invertible at $p$. 
In your case, you have to choose a generic diagonal matrix, evaluate the jacobian in it and then decide for which values it does not become singular.

More details explained. Recall that here $M(2,\mathbf{R})$ has the same Banach space structure as $\mathbf{R}^4$, the variables becoming the entries of the matrix. Your map is then written, by means of a standard choice of coordinates, as 
$$f(a,b,c,d)=(a^2+bc,b(a+d),c(a+d),bc+d^2)$$
Hence, the full jacobian matrix is
$$\mathrm{Jac}(f,A)= \left ( \begin{matrix} 2a & c & b & 0 \\ 
b & d+a & 0 & b \\ c & 0 & a+d & c \\ 0 & c & b & 2d \end{matrix} \right )$$
and when you evaluate at a diagonal matrix $D=\mathrm{diag}(d_1,d_2)$ you get
$$\mathrm{Jac}(f,D)= \left ( \begin{matrix} 2d_1 & 0 & 0 & 0 \\ 
0 & d_1+d_2 & 0 & 0 \\ 0 & 0 & d_1+d_2 & 0 \\ 0 & 0 & 0 & 2d_2 \end{matrix} \right )$$
Now, to see when this matrix is non-singular it is sufficient to determine where its determinant vanishes, namely
$$|\mathrm{Jac}(f,D)|=4d_1d_2(d_1+d_2)^2\neq 0$$
It follows, finally, that the matrices $D$ around which $f$ is a local diffeomorphism are exactly the non-singular ones ($d_1d_2\neq 0$) in which $d_1\neq -d_2$.
