Chapter 4.2, Problem 44E

(a)

To determine

To calculate: The value of $P^2=P\cdot P$ when $P=\left[\begin{array}{ccc}-0.3& -0.3& -0.3\\ 0.9& 0.9& 0.9\\ 0.4& 0.4& 0.4\end{array}\right]$.

(b)

To determine

To calculate: The value of $P^4=P^2\cdot P^2$ when $P=\left[\begin{array}{ccc}-0.3& -0.3& -0.3\\ 0.9& 0.9& 0.9\\ 0.4& 0.4& 0.4\end{array}\right]$.

(c)

To determine

To calculate: The value of $P^8$ when $P=\left[\begin{array}{ccc}-0.3& -0.3& -0.3\\ 0.9& 0.9& 0.9\\ 0.4& 0.4& 0.4\end{array}\right]$.

(d)

To determine

To calculate: The value of $P^{1000}$ without calculating explicitly when $P=\left[\begin{array}{ccc}-0.3& -0.3& -0.3\\ 0.9& 0.9& 0.9\\ 0.4& 0.4& 0.4\end{array}\right]$. Also, what do you notice about the columns of $P$? Compare with Exercise $40$, and state a general result about square matrices that these two exercises seem to suggest.