Kunze Linear Algebra · Problem 6.3.2

Elementary Canonical Forms

Mathematics — Methods · minimal-polynomial-companion-matrix

Problem

Let aa, bb, and cc be elements of a field FF, and let AA be the following 3×33 \times 3 matrix over FF: A=[00c10b01a].A = \begin{bmatrix} 0 & 0 & c \\ 1 & 0 & b \\ 0 & 1 & a \end{bmatrix}. Prove that the characteristic polynomial for AA is x3−ax2−bx−cx^3 - ax^2 - bx - c and that this is also the minimal polynomial for AA.

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