Question 21
Consider a positive integer $n$ and $A, B$ two matrices in $M_n(\mathbb{C})$ such that $A^2 + B^2 = 2AB$. Prove that:
a) The matrix $AB - BA$ is not invertible.
b) If the rank of $A - B$ is $1$, then matrices $A$ and $B$ commute.
a) The matrix $AB - BA$ is not invertible.
b) If the rank of $A - B$ is $1$, then matrices $A$ and $B$ commute.