Ok, so I am the matrix Lie group of invertible $n^2 \times n^2 $ matrices is connected. Only approach I can think of is showing it's path connected and therefore connected.
Simplest path is the Complex straight line tz+(1-t)w between invertible matrices w to z. But how can
I show determinant is non-zero throughout the path? I know $Det(AB)=DetADetB$ but I don't know how to
deal with $Det(tz +(1-t)w)$ to show it's never 0. I guess since Det is a polynomial of degree n we
could alsways tweak the path at most n times to avoid any roots we may run into. Is that enough
to show connectedness?
