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Assume that all $a_j$'s are nonzero and $B$ is an $n$-by-$n$ weighted shift matrix with weights $b_1,..., b_n$. We show that $B$ is unitarily equivalent to $A$ if and only if $b_1... b_n=a_1...a_n$ and, for some fixed $k$, $1\\le k \\le n$, $|b_j| = |a_{k+j}|$ ($a_{n+j}\\equiv a_j$) for all $j$. 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Assume that all $a_j$'s are nonzero and $B$ is an $n$-by-$n$ weighted shift matrix with weights $b_1,..., b_n$. We show that $B$ is unitarily equivalent to $A$ if and only if $b_1... b_n=a_1...a_n$ and, for some fixed $k$, $1\\le k \\le n$, $|b_j| = |a_{k+j}|$ ($a_{n+j}\\equiv a_j$) for all $j$. 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