Wikipedia says:

To look at it slightly more quantitatively consider the well-known problem:

Consider again, a random square n by n matrix A, whose entries are restricted to the set of integers {-p, -p+1, ..., 0, ... p-1, p}. Each of the 2p+1 values are equally probable.

Thus, if p = 1, then the matrix A has entries {-1, 0, 1}.

Singular matrices are rare in the sense that if you pick a random square matrix over a continuous uniform distribution on its entries, it will almost surely not be singular.This means that if you use generate a random matrix (using matlab's rand function for example) the resulting matrix is likely to be nonsingular.

*How*likely? That depends on the size of the matrix.To look at it slightly more quantitatively consider the well-known problem:

Consider an n by n square matrix A, whose entries are, with equal probability, either 0 or 1. What is the probability that this matrix is invertible (or nonsingular)?Let us consider a slightly generalized version of the problem.

Consider again, a random square n by n matrix A, whose entries are restricted to the set of integers {-p, -p+1, ..., 0, ... p-1, p}. Each of the 2p+1 values are equally probable.

Thus, if p = 1, then the matrix A has entries {-1, 0, 1}.

- What is the probability that this matrix is nonsingular?
- What is the probability that this matrix is nondefective?

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