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Give an integer n≥2, let M(n) be the n×n matrix defined as follows, where Mᵢⱼ(n) denotes the entry in row i, column j of M(n): [tex] \rm M_{ij}(n)=\begin{cases} \bigg( \rm \dfrac{j}{j - 1} \bigg)^{2} & \rm if \: i = j + 1 \: and \: j > 1\\& \rm 1 \: if \:i = 2 \: and \: j = 1 \\ & \rm0 \: otherwise\end{cases}[/tex] Define matrix A(n) as follows: [tex] \rm A(n) = \sum_{k = 0}^{ \infty } (M (n))^k \\ [/tex]
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