Subject: Applied Mathematics 4

Topic: Matrix Theory

Difficulty: Medium

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

$$ \begin{vmatrix} 2-\lambda& -2& 3 \\ 1& 1-\lambda& 1 \\ 1& 3& -1-\lambda \end{vmatrix} = 0 $$

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

$ \lambda^3 - (2)\lambda^2 + (-4-5+4)\lambda + 6 = 0 \\ \lambda^3 - 2 \lambda^2 - 5 \lambda + 6 = 0 \\ (\lambda - 1)(\lambda - 3)(\lambda + 2) = 0 $

Therefore the eigen values are 1,3,-2

The eigen values are distinct.

Therefore, A is non-derogatory.