## 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*)