Matrix/R/2x2/Characterize eigenspace configuration/Exercise

Consider the real matrices

${}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \operatorname {Mat} _{2}(\mathbb {R} )\,.$

Characterize, in dependence on ${}a,b,c,d$, when such a matrix has

1. two different eigenvalues,
2. one eigenvalue with a two-dimensional eigenspace,
3. one eigenvalue with a one-dimensional eigenspace,
4. no eigenvalue.

Create a solution