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 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 (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 .

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