Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$.

What matrices belongs to $S$, precisely?

Let $M=Det^{-1}\{0\}-S$ be the codimension one submanifold of $M_{n}(\mathbb{R})$ which has a natural Riemannian metric induced by the standard metric of $M_{n}(\mathbb{R})\simeq\mathbb{R}^{n^{2}}.$

What is a linear algebraic and matrix meaning for a matrix $A\in M$ with the following property:

"The sectional curvature of $M$ at $A$ is independent of choosing a $2$-plane tangent to $M$ at $A$"

What is a precise example of this situation, for $n=2$?

Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function.

What matrices belongs to $S$, precisely?

Let $M=Det^{-1}\{0\}-S$ be the codimension one submanifold of $M_{n}(\mathbb{R})$ which has a natural Riemannian metric induced by the standard metric of $M_{n}(\mathbb{R})\simeq\mathbb{R}^{n^{2}}.$

What is a linear algebraic and matrix meaning for a matrix $A\in M$ with the following property:

"The sectional curvature of $M$ at $A$ is independent of choosing a $2$-plane tangent to $M$ at $A$"

What is a precise example of this situation, for $n=2$?