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integer matrix BA^kB^{-1} , det (A) = 1, det (B) \ne 0

Source: 15th -b QEDMO problem 7 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

May 30, 2021

Problem Statement

Let A,BA, B be integer n×nn\times n-matrices, where det(A)=1det (A) = 1 and det(B)≠0det (B) \ne 0. Show that there is a k∈Nk \in N, for which BAkB−1BA^kB^{-1} is a matrix with integer entries.
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