Check whether the following matrix is derogatory or non-derogatory.

$A=\begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$

Solution:

$A=\begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$

$|A|=32$

Characteristic Equation $|A- \lambda I|=0$

$\begin{vmatrix} 6-\lambda & -2 & 2 \\ -2 & 3-\lambda & -1 \\ 2 & -1 & 3-\lambda \end{vmatrix}=0$

$\lambda^{3}-12\lambda^{2}+(18+9+18-4-4-1)\lambda -32=0$

$\lambda^{3}-12\lambda^{2}+36\lambda-32=0$

$\lambda=8, \ …