Consider the following Matrix, are the conditions satisfied?

Question

I am trying to solve the following questions however i am having little lucky. I have searched online and consulted my textbook however neither are proving to be helpful. These types of questions always seem to come up in exams so i would be grateful for any help! I have attached the link to the questions below, many thanks!

Answer 1

We have $$\textbf B=\textbf I-\textbf A=\begin{pmatrix}1&0&0 \\ 0&1&0 \\ 0&0&1 \end{pmatrix}-\begin{pmatrix}0.1&0.5&0.2 \\ 0&0.1&0 \\ 0&0.2&0.2 \end{pmatrix}=\begin{pmatrix}0.9&-0.5&-0.2 \\ 0&0.9&0 \\ 0&-0.2&0.8 \end{pmatrix}$$

Calculating the leading principal minors:

$|\textbf B_1| =|0.9|=0.9>0$

$|\textbf B_2|=\left|\begin{matrix}0.9&-0.5 \\ 0&0.9 \end{matrix}\right|=0.9\cdot 0.9-(0\cdot(-0.5))=0.81>0$

$|\textbf B_3|=\left|\begin{matrix}0.9&-0.5&-0.2 \\ 0&0.9&0 \\ 0&-0.2&0.8 \end{matrix}\right|=...$

If $|\textbf B_1|,|\textbf B_2|,|\textbf B_3| >0$, then the Hawkins–Simon condition is fullfilled.