Question 23
Let $m,n$ be natural numbers, $m,n\ge2$. Consider matrices $A_1,A_2,\ldots,A_m\in\mathcal M_n(\mathbb R)$ which are not all nilpotent. Prove that there is an integer $k\gt 0$ such that $A_1^k+A_2^k+\cdots+A_m^k\ne O_n$.
Note: A square matrix is called nilpotent if one of its powers is zero.
Second source statement (Hungarian version): Given natural numbers $m,n\ge2$ and matrices $A_1,A_2,\ldots,A_m\in\mathcal M_n(\mathbb R)$ which are not all nilpotent, prove that there is an integer $k\gt 0$ for which $A_1^k+A_2^k+\cdots+A_m^k\ne O_n$.
Note: A square matrix is nilpotent if one of its powers is the zero matrix.
Note: A square matrix is called nilpotent if one of its powers is zero.
Second source statement (Hungarian version): Given natural numbers $m,n\ge2$ and matrices $A_1,A_2,\ldots,A_m\in\mathcal M_n(\mathbb R)$ which are not all nilpotent, prove that there is an integer $k\gt 0$ for which $A_1^k+A_2^k+\cdots+A_m^k\ne O_n$.
Note: A square matrix is nilpotent if one of its powers is the zero matrix.