Science:Math Exam Resources/Courses/MATH307/December 2010/Question 07 (c)/Solution 1

Taking the limit n → ∞ on both sides yields

${\begin{aligned}\lim _{n\to \infty }\mathbf {y} _{n}&=\lim _{n\to \infty }(A-3I)^{-1}\mathbf {x} _{n-1}\\&=(A-3I)^{-1}\lim _{n\to \infty }\mathbf {x} _{n-1}\\&=(A-3I)^{-1}\mathbf {x} _{\infty }\\&=\eta _{3}\mathbf {x} _{\infty }\\&=\mathbf {x} _{\infty }=[1/2,1/2,1/2,1/2]^{T}\end{aligned}}$

We used, from part (a), that $\mathbf {x} _{\infty }$ is an eigenvector of $\displaystyle (A-3I)^{-1}$ with eigenvalue $\eta _{3}=1$.