If you have *n* equations in *n* unknowns over a finite field with *q* elements, how likely is it that the system of equations has a solution?

The number of possible *n* × *n* matrices with entries from a field of size *q* is *q*^{n²}. The set of invertible *n* × *n* matrices over a field with *q* elements is *GL*_{n}(*q*) and the number of elements in this set is [1]

The probability that an *n* × *n* matrix is invertible is then

which is an increasing function of *q* and a decreasing function of *n*. More on this function in the next post.

## Related posts

- Spaces and subspaces over finite fields
- Finite projective planes
- Finite field Diffie Hellman encryption

[1] Robert A. Wilson. The Finite Simple Groups. Springer 2009