two problems to finding the regular value of matrix group $1$. Let $F:M_2(\mathbb{R}) \to M_2(\mathbb{R})$ given by $F(X)=X^TX$.   
$2$.$F:M_2(\mathbb{R}) \to S_2(\mathbb{R})$ given by $F(X)=X^TX$ where $S_2(\mathbb{R})=$ {$X \in M_2(\mathbb{R}): X^T=X$}.
Does $O$ and $I$ are regular values of 1 and 2?

My thoughts:-
after calculating the derivative I get $D_vF(X)=v^tX+X^Tv$.
taking $X=I$ it becomes $D_vF=v^t+v$.
sothe range set is the set of symmetric matrix.
For 1. no value will be a regular value as the given  range is $M_2(\mathbb{R})$.   
For 2.$O$ is not a regular point since $D_OF=O$.But not sure about $I$.But I think it will be a regular value but cant find proper justification.

*

*Instead of using the space of all matrices as your target, use the space of symmetric matrices as the target. 

*Compute kernel of the derivative and it's dimension. 

*Conclude that the derivative is surjective using rank+nullity theorem. 

*Conclude that the preimage of 1 is a submanifold. 

*(Bonus) Repeat steps 1-4 for matrices of arbitrary dimension. 
