In the space of $2\times 2$ matrices, find explicitly the sets of matrices with 1)a single zero eigenvalue, 2) a pair of pure imaginary eigenvalues. Show that each set is a submanifold of $\mathbb{R}^4$ and find its codimension. (Hint: Use the Implicit Function Theorem). I thought the first set is the following \begin{equation} \left\{\left( \begin{array}{cc} a & b \\ c & d \end{array} \right)|a,b,c,d\in \mathbb{R}, \ \ ad-bc=0, \ \ a+d\neq0 \right\} \end{equation} and the second is:
\begin{equation} \left\{\left( \begin{array}{cc} a & b \\ c & d \end{array} \right)|a,b,c,d\in \mathbb{R},\ \ (a+d)^2-4(ad-bc)<0\right\} \end{equation}
I do not know how to continue. Any suggestions please?
thank you very much