Olympiad problems

2004 Nicolae Coculescu, 3

Solve in $ \mathcal{M}_2(\mathbb{R}) $ the equation $ X^3+X+2I=0. $ [i]Florian Dumitrel[/i]

Do exist pairwise distinct matrices $ A,B,C\in \mathcal{M}_2(\mathbb{R}) $ verifying the following properties? $ \text{(i)} \det A=\det C$ $ \text{(ii)} AB=C,BC=A,CA=B $ $ \text{(iii)} \text{tr} A,\text{tr} B\neq 0 $ [i]Robert Pop[/i]

2006 Grigore Moisil Urziceni, 2

Tags: linear algebra , cayley-hamilton

Let be two matrices $ A,B\in\mathcal{M}_2\left( \mathbb{C} \right) $ satisfying $ AB-BA=A. $ Show that: [b]a)[/b] $ \text{tr} (A) =\det (A) =0 $ [b]b)[/b] $ AB^nA=0, $ for any natural number $ n $

2003 Gheorghe Vranceanu, 1

Tags: matrix , linear algebra , cayley-hamilton

Prove that if a $ 2\times 2 $ complex matrix has the property that there exists a natural number $ n $ such that $ \text{tr}\left( A^n\right) =\text{tr}\left( A^{n+1} \right) =0, $ then $ A^2=0. $

2004 Nicolae Coculescu, 3

2019 Teodor Topan, 1

2006 Grigore Moisil Urziceni, 2

2003 Gheorghe Vranceanu, 1