Exercises From Cartan Subalgebras and Root Systems

Why are diagonal matrices their own normaliser?
How you do define determinant for a general linear operator? (i.e without defining a basis)
Why is the exponential map well defined for matrices? $exp: gl_{n} \rightarrow GL(g)$ Where the exponential map takes a matrix to its exponential Taylor series.
Consider $gl_{2} \& gl_{3}$ , what is $\mathrm{ad}(x)$ , and what is $e^{\mathrm{ad}(x)}$ ?
Show the rank of $gl_{n}$ is n, where rank is the dimension of a Cartan Subalgebra.
Show that the regular elements in $gl_{n}$ are diagonal matrices with no repeated entries.
If we define the map $-1 : V \rightarrow V$ which takes a root $\alpha \rightarrow -\alpha$ , For which root systems is this in the Weyl Group?
(Very optional) Why is the Dynkin diagram for $E_{8}$ not allowed to have a connection on the central root?