I have studied positive definite matrices . And I came across this exercise.
I can show that A+B is a positive definite matrix by the definition of positive definite matrix
Also in part (2) I can say AB is not possitive definite as it not necessarily symmetric
In part (3) I can conclude that A^2 is positive definite because all its eigenvalues are positive since A is positive definite
But I am not getting idea to proceed in rest of the parts