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Romania District Olympiad 2002 - Grade XI

Source:

March 18, 2011
linear algebramatrixfunctionlinear algebra unsolved

Problem Statement

a)Find a matrix A∈M3(C)A\in \mathcal{M}_3(\mathbb{C}) such that A2≠O3A^2\neq O_3 and A3=O3A^3=O_3. b)Let n,p∈{2,3}n,p\in\{2,3\}. Prove that if there is bijective function f:Mn(C)→Mp(C)f:\mathcal{M}_n(\mathbb{C})\rightarrow \mathcal{M}_p(\mathbb{C}) such that f(XY)=f(X)f(Y), ∀X,Y∈Mn(C)f(XY)=f(X)f(Y),\ \forall X,Y\in \mathcal{M}_n(\mathbb{C}), then n=pn=p.
Ion Savu
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