You are applying three invertible linear transformations to your ellipse, so I'm sure you are right that the shape is still an ellipse. You should be able to prove this by finding the equation of the new shape. That should be routine, if messy: take a point on the new shape, apply the inverse of the three transformations you applied to the starting ellipse, now you have a point on the old ellipse, so you can plug the new point into the equation of the first ellipse.

Unless you want an explicit equation for the new shape you should be able to convince yourself the shape is still an ellipse by reasoning as follows: Any equation of the form

axx+bxy+cyy+dx+ey+f=0 is the equation of one of the following shapes : circle,ellipse,parabola,hyperbola, straight line, a pair of straight lines, or possibly 0,1 or 2 points. The substitution I am suggesting to find the new equation is linear in x and y, so you will end up with an equation of the same form. But clearly invertible linear transformations cannot turn an ellipse into any of the other possibilities apart from a circle. So applying invertible linear transformations to an ellipse you can only finish up with another ellipse or a circle.