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I-a^p

Source: Romanian MO 2006, District Round

March 11, 2006
linear algebramatrixinequalitiesalgebrapolynomialabstract algebralinear algebra unsolved

Problem Statement

Let n,p≥2n,p \geq 2 be two integers and AA an n×nn\times n matrix with real elements such that Ap+1=AA^{p+1} = A. a) Prove that rank(A)+rank(In−Ap)=n\textrm{rank} \left( A \right) + \textrm{rank} \left( I_n - A^p \right) = n. b) Prove that if pp is prime then rank(In−A)=rank(In−A2)=…=rank(In−Ap−1). \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) .
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