Determinant zero 2 by 2 Matrices

Random number: 91

20th April 2017

So apparently, $GL(n,\mathbb{R})$, the set of invertible (determinant is not equal to zero) matrices forms a manifold in $n^2$ Euclidean space (with each element of the matrix associated with a coordinate). That's pretty amazing right! So to understand this manifold, I thought I'd consider it's complement, (determinant zero matrices - this is actually an algebraic variety?) in the simplest case ($n$ = 2).

Label the matrices: $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ Now, we can look at cross-sections of this variety by fixing a particular value of $a$. Here are my bad drawings of this: