Subject: Applied Mathematics 4

Topic: Matrix Theory

Difficulty: Medium

For characteristic equation |A - $\lambda$ I| = 0

$$ \begin{vmatrix} 7 -\lambda& 4&-1 \\ 4& 7-\lambda&-1 \\ -4&-4& 4-\lambda \end{vmatrix} = 0 $$

$\lambda^3$ - (Sum of diagonal elements)$\lambda^2$ + (Sum of Principal minors)$\lambda$ - |A| = 0

$ \lambda^3 - (7+7+4)\lambda^2 + (24+24+33)\lambda - 108 = 0 \\ \lambda^3 - 81 \lambda^2 + 81 \lambda - 108 = 0 \\ (\lambda - 3)(\lambda - 3)(\lambda - 12) = 0 \\ \therefore \lambda = 3,3,12 $

The minimal polynomial is $ (\lambda - 3)(\lambda - 12) = 0 \implies \lambda^2 - 15 \lambda + 36 = 0 $

The degree of minimal polynomial is less than the degree of characteristic polynomial.

Therefore, A is derogatory.