Subject: Applied Mathematics 2

Topic: Matrices

Difficulty: Medium

Matrix A is said to be non derogatory if the degree of minimal polynomial is same as the order of matrix A. since the roots of minimal polynomial are same as the roots of characteristic equation i.e λ.

First find eigen values.

To find eigen values of A ,characteristic eqation is $|A- λI| =0$

i.e. $ \begin{vmatrix} 2-λ&-2&3 \\ 1&1-λ&1 \\ 1&3&-1-λ \end{vmatrix} $ = 0 , expanding we get a cubic equation in λ

as $λ^3 - λ^2 - 7λ + 6 = 0$, $i.e. λ = 1, - 2, 3 $

Let $f(x)$ be a minimal polynomial of matrix A . Since the roots of minimal polynomial is same as roots of characteristic eqn $f(x)$ will have roots 1, -2, 3 i.e. factors are $(x-1), (x+2) \& (x-3)$

Since $f(x) = ( x-1)(x+2)(x-3) = x^3 - x^2 – 7x +6$ annihilates A

i.e. $f(A) = A3 – A2 – 7A +6 = $ $ \begin{bmatrix} 0&0&0 \\ 0&0&0 \\ 0&0&0 \end{bmatrix} $

Since $f(x)$ is a polynomial of lowest degree which annihilates A ,therefore $f(x)$ is a minimal polynomial . Since the degree of minimal polynomial is 3 ,equals to order of matrix, matrix A is non derogatory.