I am having trouble with this question and need some help.

Let $\displaystyle A$ be a square matrix of order $\displaystyle n$ over $\displaystyle F$ with a rank of 1.

1. Show that the characteristic polynomial of $\displaystyle A$ is $\displaystyle P(t)=t^{n}-tr(A)t^{n-1}$ and find the minimal polynomial of $\displaystyle A$.

2. Let $\displaystyle T:F^{n}\rightarrow F^{n}$ be the linear map defined by $\displaystyle Tv=Av$ for every $\displaystyle v\in F^{n}$. Show that for every $\displaystyle 1\leq k\leq n$ there exists a subspace of $\displaystyle F^{n}$ of dimension $\displaystyle k$ that is $\displaystyle T-$reserved.

I can show the first part of 1. If the rank is 1 then the nullity of $\displaystyle A$ is of dimension $\displaystyle n-1$ and therfore the geometric multiplicity of the eignvalue $\displaystyle 0$ is $\displaystyle n-1$, so we get that the algebraic multiplicity of $\displaystyle 0$ is at least $\displaystyle n-1$. Therfore the characteristic polynomial is of the form $\displaystyle t^{n-1}(t-a)=t^{n}-at^{n-1}$ (where $\displaystyle a$ is the other eignvalue). We have shown before in a previous lin alg class that the coefficient of $\displaystyle t^{n-1}$ of the characteristic polynomial of any square matrix of order $\displaystyle n$ is $\displaystyle -tr(A)$, so we get then that the characteristic polynomial is indeed $\displaystyle t^{n}-tr(A)t^{n-1}$.

Now for the minimal polynomial I know that it has to be of the form $\displaystyle t^{k}(t-tr(A))$ where $\displaystyle k\leq n-1$, but I am not sure how to proceed.

As for 2 I would appreciate some direction.

Thanks,

SK