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Found problems: 2

2016 Korea USCM, 8
For a $n\times n$ complex valued matrix $A$, show that the following two conditions are equivalent. (i) There exists a $n\times n$ complex valued matrix $B$ such that $AB-BA=A$. (ii) There exists a positive integer $k$ such that $A^k = O$. ($O$ is the zero matrix.)
nilpotent matrix commutator linear algebra matrices number theory

2016 Korea USCM, 6
$A$ and $B$ are $2\times 2$ real valued matrices satisfying $$\det A = \det B = 1,\quad \text{tr}(A)>2,\quad \text{tr}(B)>2,\quad \text{tr}(ABA^{-1}B^{-1}) = 2$$ Prove that $A$ and $B$ have a common eigenvector.
matrix commutator trace linear algebra common eigenvector determinant determinants matrices