Two square matrices A and B of the same size do not commute.Prove that AB and BA has the same set of eigenvalues.

I did in the following way:Please check if I am correct.
Consider: det(AB-yI)*det(A) where y represents eigenvalues and
I represents unit matrix
=det[(AB-yI)A]
=det[(AB)A-(yI)A]
=det[A(BA)-A(yI)]
=det(A)*det(BA-yI)
det(A) is not equal to zero,in general.
So,if det(AB-yI)=0,det(BA-yI)=0 also.
hence, conclusion.

Your proof is correct.

Another cute method:
AB(x)=y(x)
Pre-multiply with B
BAB(x)=yB(x)
(BA)[Bx]=y[Bx]
hence,proved.