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## Homework Statement

Let A = [itex]\begin{pmatrix}1 & 1 & 0 & 0\\-1 & -1 & 0 & 0\\-2 & -2 & 2 & 1\\ 1 & 1 & -1 & 0 \end{pmatrix}[/itex]

The characteristic polynomial is [itex]f(x)=x^2(x-1)^2[/itex]. Show that f(x) is also the minimal polynomial of A.

Method 1: Find v having degree 4.

Method 2: Find a vector v of degree 2, whose minimal polynomial (on A of v) is [itex]x^2[/itex], and another, w, whose minimal polynomial is [itex](w-1)^2[/itex]. Or, just show that v and w exist.

## The Attempt at a Solution

I'm confused as to how a vector can have a degree of more than 1. Isn't a vector simply:

[itex]v= \begin{pmatrix}a\\b\\c\\...\\n\end{pmatrix}[/itex] in [itex]R^n[/itex]? I think I can get the question once I understand this. Thanks!