Automated Matrix Multiplication

\[ A = \begin{bmatrix} 560 & 532 & 442 & 389 \\ 51 & 690 & 478 & 12 \\ 209 & 263 & 155 & 608 \\ 89 & 466 & 730 & 835 \end{bmatrix}\]

Consider the matrix above.

Let \(S\) be the sum of all the entries in \(A^{100}\). Find the sum of digits of \(S\)

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