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Let $A$ be a square matrix. In the following, a sequence of matrices $C_{j}$ converges if all of its elements converge.
Prove that the following are equivalent:
(i) $A^{k} \rightarrow 0$ as $k \rightarrow \infty$ [each of the elements of $A^{k}$ converge to zero].
(ii) All the eigenvalues $\lambda_{j}$ of $A$ have $\left|\lambda_{j}\right|<1$.
(iii) The matrix geometric series $\sum_{0}^{\infty} A^{k}$ converges to $(I-A)^{-1}$.
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