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Post Undeleted by Angelo
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Angelo
Angelo
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As you say, $\sum_A A^k$ is a scalar matrix. If $B$ is invertible we have $B^k\sum_A A^k = \sum_A A^k$; hence if there exists an invertible matrix whose order does not divide $k$ we can conclude that $\sum_A A^k = 0$. For example, it is not hard to show that in $M_p(\mathbb F_p)$ there are invertible matrices of order $p$ and also of order $p^p-1$; hence we can conclude that if $k$ is not a multiple of $p(p^p-1)$ then $\sum_A A^k = 0$. This does not solve the general caseMy answer was nonsense, but it is certainly enough to conclude for $p = 17$ and $k = 80$sorry.

As you say, $\sum_A A^k$ is a scalar matrix. If $B$ is invertible we have $B^k\sum_A A^k = \sum_A A^k$; hence if there exists an invertible matrix whose order does not divide $k$ we can conclude that $\sum_A A^k = 0$. For example, it is not hard to show that in $M_p(\mathbb F_p)$ there are invertible matrices of order $p$ and also of order $p^p-1$; hence we can conclude that if $k$ is not a multiple of $p(p^p-1)$ then $\sum_A A^k = 0$. This does not solve the general case, but it is certainly enough to conclude for $p = 17$ and $k = 80$.

My answer was nonsense, sorry.

Post Deleted by Angelo
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Angelo
Angelo
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As you say, $\sum_A A^k$ is a scalar matrix. If $B$ is invertible we have $B^k\sum_A A^k = \sum_A A^k$; hence if there exists an invertible matrix whose order does not divide $k$ we can conclude that $\sum_A A^k = 0$. For example, it is not hard to show that in $M_p(\mathbb F_p)$ there are invertible matrices of order $p$ and also of order $p^p-1$; hence we can conclude that if $k$ is not a multiple of $p(p^p-1)$ then $\sum_A A^k = 0$. This does not solve the general case, but it is certainly enough to conclude for $p = 17$ and $k = 80$.