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3. Let

f(x)= x^n +a_1 x^(n-1)+ \dots + a_n

(

n\geq 1

) be an irreducible polynomial over the field

K

. Show that every non-zero matrix commuting with the matrix

\begin{bmatrix} 0 & 1 & 0 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & \dots & 0 & 1 \\ -a_n & -a_{n-1} & -a_{n-2} & \dots & -a_2 & -a_1 </br>\end{bmatrix}

is invertible. (A. 4)

Problem Statement