Let $n\times n$ complex matrices $A=(a_{ij})_{1\leq i,j\leq n}$ and $B=(b_{ij})_{1\leq i,j\leq n}$ be a tri-diagonal matrix whose off-diagonal entries are non-zero and a diagonal matrix, respectively. Equivalently, the entries $a_{ij},b_{ij}\in \mathbb{C}$ of $A$ and $B$ satisfy $a_{ij}=0\ (|i-j|>1)$, $a_{ij}\neq 0\ (|i-j|=1)$, $b_{ij}=0\ (|i-j|\geq 1)$.
Suppose that $A$ and $B$ can be converted into a diagonal matrix and a tri-diagonal matrix whose off-diagonal entries are non-zero, respectively, by a similarity transformation with a common regular matrix $P:\ A\mapsto P^{-1}AP$, $B\mapsto P^{-1}BP$.
Let $\lambda_i$ be the eigenvalues of A. Hereafter, $I$ denotes the identity matrix, and $O$ denotes the zero matrix. Answer the following questions.
i.) Let $c_0,c_1,...,c_{n-1}$ be constants. Show that
$\sum_{k=0}^{n-1} c_k A^k=O$
holds only when $c_0=c_1=\cdots =c_{n-1}=0$.
ii.) Show that all the eigenvalues of $A$ are mutually distinct.
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