# On the Convergence of the Sequence of Powers of a $$2 \times 2$$ Matrix

by Roman W. Wong

Mathematics Magazine
June, 1996

Subject classification(s): Linear Algebra | Eigenvalues and Eigenvectors | Statistics and Probability | Probability | Stochastic Processes
Applicable Course(s): 3.8 Linear/Matrix Algebra | 7.3 Stochastic Processes

The fact that the limit of the $$n$$-th power of a $$2\times 2$$ matrix $$A$$ tends to $$0$$ if  $$\det A < 1$$ and $$\mid 1 + \det(A) \mid > \mid$$ tr$$(A) \mid$$ is used to prove a well-known theorem in Markov chains for $$2 \times 2$$ regular stochastic matrices and to obtain an explicit formula for the stationary matrix and eigenvector.

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