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Necessary condition for symmetric matrix

Source: SEEMOUS 2024, Problem 4

April 16, 2024
linear algebracollege contestsmatrix

Problem Statement

Let n∈Nn\in\mathbb{N}, n≥2n\geq 2. Find all values of k∈Nk\in\mathbb{N}, k≥1k\geq 1, for which the following statement holds: "If A∈Mn(C) is such that AkA∗=A, then A=A∗."\text{"If }A\in\mathcal{M}_n(\mathbb{C})\text{ is such that }A^kA^*=A\text{, then }A=A^*\text{."} (here, A∗A^* denotes the conjugate transpose of AA).
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