concept of image rotation
When you multiply the image matrix with the rotation matrix, you are essentially multiplying an integer matrix with a real number matrix. The result is a real matrix, which has to be quantized back to a discrete matrix. 

What is being quantized? The coordinates themselves. The integer coordinates of the previous position of the pixel is now a real valued coordinate, which must be converted back to an integer in order to be placed on the screen.

If you were asked to rotate it back to the original angle you could have been right in questioning the futility of interpolation. However, the user might not want to rotate it back, so you have to perform interpolation or quantization to be able to use the rotated image. If you are concerned about image fidelity, you might want to save a copy of the original real valued matrix in memory or file until or opertions are finished and the user exits program.

Here is a useful link you might want to peruse.