Thread: Probability for existence

What is the probability that a simultaneous equation has a unique solution. That is, its determinant is not-zero?

It seems that since there are infinitely many real numbers the probaility is 100%. But is that true? I also believe that if it is possible to find the probability it will change depending on the number of simultaneous equations you are using.

The condition that an n-by-n set of simultaneous linear equations over the reals have a unique solution is just that the determinant of the matrix of coefficients be zero. Since determinant is a polynomial in the entries of the matrix, the locus of determinant equal to zero is a set of measure zero.

The situation changes over a finite field. For example, over the field of two elements, the probability of determinant equal to zero is asymptotic to about 0.71.