GATE General Aptitude and CS Solutions Practice Test

Eigenvalues tell you how the matrix scales vectors that lie along directions which remain in the same direction after the transformation. If Ax = λx for a nonzero x, the transformation simply stretches or squeezes x by a factor λ (and may flip its direction if λ is negative). So the eigenvalue is the scaling factor for that eigenvector direction, which is exactly what the correct answer describes.

Rotation angles aren’t given by eigenvalues—the eigenstructure doesn’t generally encode how much a transformation rotates space. The number of fixed points isn’t what eigenvalues measure, and rank concerns how many independent columns there are, not how the transformation scales specific directions.