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Prove that A^p \neq I_p

Source: 2022 3rd OMpD LU P2 - Brazil - Olimpíada Matemáticos por Diversão

July 8, 2023
linear algebraalgebramatrixMatrix algebratrace

Problem Statement

Let p≥3p \geq 3 be a prime number and let AA be a matrix of order pp with complex entries. Assume that Tr(A)=0\text{Tr}(A) = 0 and det⁡(A−Ip)≠0\det(A - I_p) \neq 0. Prove that Ap≠IpA^p \neq I_p.
Note: Tr(A)\text{Tr}(A) is the sum of the main diagonal elements of AA and IpI_p is the identity matrix of order pp.
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