Prove that $2\times2$ complex matrices of determinant 0 and non null trace is connected.
What I did:
My intuition is that such matrices are similar to \begin{pmatrix} z & 0 \\ 0 & 0 \end{pmatrix} with $z\ne 0$
So $E$ is isomorphic to $GL_2(\Bbb C) \times \Bbb C^*$ therefore $E$ is connected.
Is that correct?
In $\mathbb{C}$. a matrix is similar to a triangular matrix. Therefore here, all matrices are similar to \begin{pmatrix} z & 1\\ 0 & x \end{pmatrix} with $z,x \in \mathbb{C}$.
The determinant is zero but not the trace, so the matrix is similar to \begin{pmatrix} z & 1 \\ 0 & 0 \end{pmatrix} with $z\ne 0$.
It characteristic polynomial is : $P(T) = T(T-z)$ It has two differents roots so your matrix is diagonalizable... and similar to \begin{pmatrix} z & 0 \\ 0 & 0 \end{pmatrix} with $z\ne 0$