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Consider the $2$-by-$2$ matrix $A = \begin{pmatrix} 5 & 1 \\ -1 & 3 \end{pmatrix}.$ What is the trace of $A^{100}?$

$$ Note: The trace of a $2$-by-$2$ matrix is defined as $\text{Tr}\begin{pmatrix} a & b \\ c& d \end{pmatrix} = a+d\; .$

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