Elementary matrices/Minimal polynomial/Exercise

We discuss the minimal polynomials of the elementary matrices.

a) Show that the minimal polynomial of an exchange matrix ${}V_{ij}$ is ${}X^{2}-1$.

b) Show that the minimal polynomial of a scaling elementary matrix ${}S_{k}(s)$ with ${}s\neq 1$ is

$X^{2}-(s+1)X+s.$

c) Show that the minimal polynomial of an addition matrix ${}A_{ij}(a)$ is of the form

$(X-1)^{k}.$

What is ${}k$?

Create a solution