Let $A$ be a $10\times 10$ matrix with complex entries such that all its eigenvalues are non-negative real numbers, and at least one eigenvalue is positive. Which of the following statements is always false ?
A. There exists a matrix $B$ such that $AB – BA = B$.
B. There exists a matrix $B$ such that $AB – BA = A$.
C. There exists a matrix $B$ such that $AB + BA = A$.
D. There exists a matrix $B$ such that $AB + BA = B$.
I just need a hint to start thinking.
Best Answer
B is always false. The trace of $AB-BA$ is always zero, but the trace of $A$, which is the sum of its eigenvalues, is positive.