Subject: Applied Mathematics 2

Topic: Matrices

Difficulty: Medium

Matrix A will be derogatory if the degree of minimal polynomial will be less than 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} 7-λ&4&-1 \\ 4&7-λ&-1 \\ -4&-4&4-λ \end{vmatrix} $ = 0 , expanding we get a cubic equation in λ

as $λ^3 - 18 λ^2 + 81λ -106 = 0$ i.e. λ = 3, 3, 12

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 two roots 3 ,12 i.e. factors are (x-3) & (x-12) .

Now we see whether $f(x) = (x-3) (x-12)= x^2 – 15x +36$ annihilates A

i.e.
$f(A) = A2 -15A +36I$

= $ \begin{bmatrix} 69& 60&-15 \\ 60& 69&-15 \\ -60&-60& 24 \end{bmatrix} - 15 \begin{bmatrix} 7&4&-1 \\ 4&7&-1 \\ -4&-4&4 \end{bmatrix} +36 \begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix} $

= $ \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 2 < 3 , order of matrix .Matrix A is derogatory.