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.



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> What matrices belongs to $S$, precisely?
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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 $M\_{n}(\mathbb{R})\simeq\mathbb{R}^{n^{2}}.$



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> What is a linear algebraic and matrix meaning for a matrix $A\in M$ with the following property:
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**"The sectional curvature of $M$ at $A$ is independent of choosing a $2$-plane tangent to $M$ at $A$"**



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> What is a precise example of this situation, for $n=2$?
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A geometric property of singular matrices
Your first question has an easy answer. The differential of $\det$ is
$$\sum\_{i,j}\hat a\_{ij}{\rm d}a\_{ij},$$
where $\hat A$ is the cofactor matrix. Thus a singular point is such that $\hat A=0\_n$, in other words, it has rank $\le n-2$.


Actually, ${\bf M}\_n({\mathbb R})$ can be stratified by the sets $R\_0,\ldots,R\_n$ of matrices of rank $k=0,\ldots n$ respectively. Each $R\_k$ is a submanifold of dimension $k(2n-k)$. $R\_k$ is homogeneous, in the sense that ${\bf GL}\_n({\mathbb R})\times{\bf GL}\_n({\mathbb R})$ operates transitively on it by $(P,Q)\cdot A=PAQ^{-1}$. The relative boundary of $R\_k$ is $R\_0\cup\cdots\cup R\_{k-1}$. In particular, there is no removable singularity.


