Determinant Probability Distribution

When a matrix is filled with random numbers, its determinant becomes a random variable. The determinant probability distribution answers: for a randomly generated square matrix, how is the value of $\det$ distributed — which values are likely, and how often is the matrix singular ($\det = 0$)?

The Idea

The determinant collapses a whole matrix into one number measuring the "signed area/volume" spanned by its rows. For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ it is $ad - bc$. If the entries are random, $ad - bc$ takes different values with different probabilities, and listing those probabilities is the distribution.

A Concrete Example

Place the numbers $1, 2, 3, 4$ (each used once) into the four cells at random — all $4! = 24$ arrangements equally likely — and compute $\det = ad - bc$. The main-diagonal pair $\{a,d\}$ and anti-diagonal pair $\{b,c\}$ are always complementary $2$-element subsets of $\{1,2,3,4\}$, whose products are

So $\det$ can only equal $\pm(12-2) = \pm 10$, $\pm(8-3) = \pm 5$, or $\pm(6-4) = \pm 2$. Each of these six values comes from exactly $4$ of the $24$ arrangements:

Discrete vs Continuous Entries

The shape depends entirely on how the entries are generated:

• Discrete entries (drawn from a finite set, as above) give a discrete distribution over finitely many determinant values. …