I have met this problem while preparing for the olympiad and I can't crack it:
Find the matrices $A \in \mathbb{M}_n \ (\mathbb {C})$ such that there exists a $b \in \mathbb{C}$ such that $ABA = bA$ for every $B \in \mathbb{M}_n \ (\mathbb{C})$.
Of course, $O_n$ is a solution with $b = 0$. I applied the determinant: $\det^2 (A) \ \det (B) = b^n \det (A)$. Since $b$ is fixed and $B$ is a variable, the only possible way is for $\det (A) = 0$.
Now, for $B = I_n$, $A^2 = bA$ and using induction it is clear that $A^n = b^{n-1}A$. This is all I could get. Can you please help me? But please, try not to use very advanced stuff, as this problem should be solvable during a contest. With that said, every solution is welcomed.
Take $2B$ for $B$. You get $2(ABA)=2bA=bA$. So $bA=0$. Hence either $b=0$, so $A=0$ (as you proved), or $A=0$. Thus $A=0$ in all cases.
A more interesting problem would be if you allow $b$ depend on $B$.