Find the Limit of a Matrix
Problem 50
Let
\[A=\begin{bmatrix}
\frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\
\frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\
\frac{3}{7} & \frac{3}{7} & \frac{1}{7}
\end{bmatrix}\]
be $3 \times 3$ matrix. Find
\[\lim_{n \to \infty} A^n.\]
(Nagoya University Linear Algebra Exam)
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