Exercise 22

Prove that for any $2\!\times\!2$ matrix $T$ there are scalars $c_{0},\dots,c_{4}$ that are not all $0$ such that the combination $c_{4}T^{4}+c_{3}T^{3}+c_{2}T^{2}+c_{1}T+c_{0}I$ is the zero matrix (where $I$ is the $2\!\times\!2$ identity matrix, with $1$’s in its $1,1$ and $2,2$ entries and zeroes elsewhere; see Exercise 21).

Let $p(x)$ be a polynomial $p(x)=c_{n}x^{n}+\dots+c_{1}x+c_{0}$. If $T$ is a square matrix we define $p(T)$ to be the matrix $c_{n}T^{n}+\dots+c_{1}T+c_{0}I$ (where $I$ is the appropriately-sized identity matrix). Prove that for any square matrix there is a polynomial such that $p(T)$ is the zero matrix.

The minimal polynomial $m(x)$ of a square matrix is the polynomial of least degree, and with leading coefficient $1$, such that $m(T)$ is the zero matrix. Find the minimal polynomial of this matrix.

(This is the representation with respect to ${\cal E}_{2},{\cal E}_{2}$, the standard basis, of a rotation through $\pi/6$ radians counterclockwise.)