An example: Determinants of $2\times2$ digit matrices

As an artificial but cute example, consider the determinants of $2\times2$matrices $\left[\matrix{a & b\cr c & d\cr}\right]$ where each entry is a non-negative integer in the range $0,1,\ldots,9$, that is a digit.

The problem is to find the determinants, ad - bc , of all possible matrices of this form and represent the frequency with which each value occurs as a high density plot. This amounts to finding the probability distribution of the determinant if each digit is chosen independently and uniformly at random.

A neat way of doing this uses the outer() function twice:

> d <- outer(0:9, 0:9)
> fr <- table(outer(d, d, "-"))
> plot(as.numeric(names(fr)), fr, type="h",
        xlab="Determinant", ylab="Frequency")

Notice the coercion of the names attribute of the frequency table to numeric in order to recover the range of the determinant values. The ``obvious'' way of doing this problem with for-loops, to be discussed in §, is so inefficient as to be impractical.

It is also perhaps surprising that about 1 in 20 such matrices is singular.