As Matthew Daws pointed out with his link above, Terry Tao answers your specific question. However, you might be interested in the larger framework of random matrix theory, which is includes the study of eigenvalue distributions of large random matrices. This subject makes precise Douglas Zare's comment:
I would guess that there will be some properties of the eigenvalues which hold for at least 98% of the alterations of a large matrices.
By putting special probability distributions on the space of $N \times N$ matrices, one can do everything explicitly using orthogonal polynomials, Riemann-Hilbert theory, and a whole slew of other exact tools. For example, the Gaussian Unitary Ensemble (GUE) is the class of complex $N \times N$ matrices with i.i.d. Gaussian entries constrained so that the matrix is Hermitian. Here, the distribution of largest eigenvalue follows the Tracy-Widom law, the cumulative distribution of eigenvalues is Wigner's semicircle law, and one can calculate explicitly pretty much any other quantity of interest.
Just as the central limit theorem holds for a much wider class of i.i.d. random variables than just Gaussians, much current work in the analysis of random matrix theory is to show that these properties are universal, and don't rely on the specific Gaussian structure.