Rank and characteristic/minimal Polynomial
I am having trouble with this question and need some help.
Let 
be a square matrix of order 
over 
with a rank of 1.
1. Show that the characteristic polynomial of is =t^{n}-tr(A)t^{n-1})
and find the minimal polynomial of .
2. Let 
be the linear map defined by 
for every 
. Show that for every 
there exists a subspace of 
of dimension 
that is 
reserved.
I can show the first part of 1. If the rank is 1 then the nullity of is of dimension 
and therfore the geometric multiplicity of the eignvalue 
is , so we get that the algebraic multiplicity of is at least . Therfore the characteristic polynomial is of the form =t^{n}-at^{n-1})
(where 
is the other eignvalue). We have shown before in a previous lin alg class that the coefficient of 
of the characteristic polynomial of any square matrix of order is )
, so we get then that the characteristic polynomial is indeed t^{n-1})
.
Now for the minimal polynomial I know that it has to be of the form ))
where 
, but I am not sure how to proceed.
As for 2 I would appreciate some direction.
Thanks,
SK
