Question 48
Let $n\geq2$ and let $A,B\in M_n(\mathbb R)$. Suppose there exists
$$
x\in\mathbb R\setminus\left\{0,\frac12,1\right\}
$$
such that
$$
xAB+(1-x)BA=I_n.
$$
Prove that $(AB-BA)^n=O_n$, where $I_n$ and $O_n$ denote the identity and zero matrices, respectively.
$$
x\in\mathbb R\setminus\left\{0,\frac12,1\right\}
$$
such that
$$
xAB+(1-x)BA=I_n.
$$
Prove that $(AB-BA)^n=O_n$, where $I_n$ and $O_n$ denote the identity and zero matrices, respectively.