Chapter 9.3, Problem 60E

Refer to the matrices $P$ and $Q$ of Problem 59. For $k$ a positive integer, let $T_k=I+Q+Q^2+\cdots +Q^k$.

(A) Explain why $T_{k+1}=T_{k}Q+I$.

(B) Using a graphing calculator and part (A) to quickly compute the matrices $T_k$, discover and describe the connection between ${\left(I-Q\right)}^{-1}$ and $T_k$ when $k$ is large.