Let $A$ be a $10 \times 10$ matrix with complex entries s.t. all eigenvalues are non negative real and at least one eigenvalue is positive. Then which of the following statements is always false?
A. there is a matrix $B$ s.t. $AB-BA=B$
B.there is a matrix $B$ s.t. $AB-BA=A$
C.there is a matrix $B$ s.t. $AB+BA=A$
D.there is a matrix $B$ s.t. $AB+BA=B$
I am new comer in Liner algebra. I have studied finite dimensional vector space, eigen value eigen vector from a book of G. Strang. I have found this in a competitive exam. I have no idea how to tackle this question. Can anybody help to solve this problem.