Distribution of Determinants of 3×3 Matrices from {1,2,…,9}

R
combinatorics
linear algebra
simulation
An exact combinatorial result and Monte Carlo approximation for determinants of all permutation matrices of the digits 1–9
Author

Michael Friendly

Published

May 1, 2026

Introduction

From the time, long ago, I first studied linear algebra, I’ve always had a fascination with the idea of the determinant of a square matrix, \(\text{det}(\mathbf{A}) \equiv | \mathbf{A} |\). You just take your matrix \(\mathbf{A}\), which could be \((2 \times 2)\), or \((10 \times 10)\), or \((100 \times 100)\), … and \(| \mathbf{A} |\) turns that into a single number, that expresses many of it’s properties—algebraic, geometric and statistical.

In short, the determinant \(| \mathbf{A} |\) is a measure of the “size” of \(\mathbf{A}\).

This post explores an interesting combinatorial and statistical problem: What is the distribution of determinants of all 3×3 matrices that can be formed from the numbers 1 through 9, each used exactly once? It arose from a simpler question Math StackExchange: What is the total number 3x3 matrices using only digits 1 to 9?

We answer this question in two ways:

  1. Exact solution: Calculate the determinant for all 9! = 362,880 permutations
  2. Permutation distribution: Generate a large number of random permutations and approximate the distribution

Setup

Code 
library(ggplot2)
library(dplyr)
library(arrangements)
library(moments)

Exact Solution: All 9! Permutations

\(9! = 362,880\) is a large number, but no so large to preclude an elegant two-line computational solution, even if it takes time and storage space.

  • Use arrangements::permutations() to generate all of these
  • Use apply() to calculate their determinants, with a helper function det_from_vec()

Computing exact distribution using all 362,880 permutations of the digits 1-9.

Code 
det_from_vec <- function(v) {
  M <- matrix(v, nrow = 3, ncol = 3, byrow = TRUE)
  det(M)
}

all_perms <- arrangements::permutations(1:9, k = 9)
determinants_exact <- apply(all_perms, 1, det_from_vec)

Random Permutation Distribution

Generating a Monte Carlo approximation with a large random sample of permutations.

Code 
B <- 100000
set.seed(42)

determinants_random <- replicate(B, {
  v <- sample(1:9, 9, replace = FALSE)
  det_from_vec(v)
})

Summary Statistics

Exact Solution

Code 
if (!is.null(determinants_exact)) {
  cat("EXACT SOLUTION (all 9! permutations):\n")
  cat(sprintf("  Total permutations: %d\n", length(determinants_exact)))
  cat(sprintf("  Range: [%.1f, %.1f]\n",
              min(determinants_exact), max(determinants_exact)))
  cat(sprintf("  Mean: %.2f\n", mean(determinants_exact)))
  cat(sprintf("  Median: %.2f\n", median(determinants_exact)))
  cat(sprintf("  SD: %.2f\n", sd(determinants_exact)))
  cat(sprintf("  Unique values: %d\n", length(unique(determinants_exact))))
  n_zero <- sum(determinants_exact == 0)
  cat(sprintf("  Zero determinants: %d (%.2f%%)\n",
              n_zero, 100 * n_zero / length(determinants_exact)))
}
EXACT SOLUTION (all 9! permutations):
  Total permutations: 362880
  Range: [-412.0, 412.0]
  Mean: -0.00
  Median: 0.00
  SD: 154.85
  Unique values: 3891
  Zero determinants: 1488 (0.41%)

Random Permutation Sample

Code 
cat("RANDOM PERMUTATION SAMPLE (B =", B, "):\n")
RANDOM PERMUTATION SAMPLE (B = 1e+05 ):
Code 
cat(sprintf("  Range: [%.1f, %.1f]\n",
            min(determinants_random), max(determinants_random)))
  Range: [-412.0, 412.0]
Code 
cat(sprintf("  Mean: %.2f\n", mean(determinants_random)))
  Mean: 0.05
Code 
cat(sprintf("  Median: %.2f\n", median(determinants_random)))
  Median: 0.00
Code 
cat(sprintf("  SD: %.2f\n", sd(determinants_random)))
  SD: 155.43
Code 
cat(sprintf("  Unique values: %d\n", length(unique(determinants_random))))
  Unique values: 3451

Visualizations

Comparison of Both Methods

Code 
if (!is.null(determinants_exact)) {
  df_combined <- rbind(
    data.frame(determinant = determinants_exact,
               method = "Exact (all 9! permutations)"),
    data.frame(determinant = determinants_random,
               method = paste0("Random sample (B = ", format(B, big.mark = ","), ")"))
  )

  ggplot(df_combined, aes(x = determinant, fill = method)) +
    geom_histogram(aes(y = after_stat(density)),
                   bins = 100, alpha = 0.6, position = "identity") +
    geom_density(aes(color = method), linewidth = 1, fill = NA) +
    scale_fill_manual(values = c("#3498db", "#e74c3c")) +
    scale_color_manual(values = c("#2c3e50", "#c0392b")) +
    labs(
      title = "Distribution of Determinants of 3×3 Matrices from {1,2,...,9}",
      subtitle = "Each matrix uses digits 1-9 exactly once",
      x = "Determinant", y = "Density", fill = "Method", color = "Method"
    ) +
    theme_minimal(base_size = 12) +
    theme(legend.position = "bottom",
          plot.title = element_text(face = "bold", size = 14),
          panel.grid.minor = element_blank())
}

Exact Distribution (Detailed View)

Code 
if (!is.null(determinants_exact)) {
  ggplot(data.frame(determinant = determinants_exact), aes(x = determinant)) +
    geom_histogram(aes(y = after_stat(density)),
                   bins = 120, fill = "#3498db", alpha = 0.7) +
    geom_density(color = "#2c3e50", linewidth = 1.2) +
    labs(
      title = "Exact Distribution: All 362,880 Permutations",
      subtitle = "Determinants of 3×3 matrices using digits 1-9 once each",
      x = "Determinant", y = "Density"
    ) +
    theme_minimal(base_size = 12) +
    theme(plot.title = element_text(face = "bold", size = 14),
          panel.grid.minor = element_blank())
}

Additional Analysis

Quartiles

Code 
if (!is.null(determinants_exact)) {
  cat("Quartiles (exact):\n")
  print(quantile(determinants_exact, probs = c(0, 0.25, 0.5, 0.75, 1)))
}
Quartiles (exact):
  0%  25%  50%  75% 100% 
-412 -101    0  101  412

Most Common Values

Code 
if (!is.null(determinants_exact)) {
  cat("Top 10 most frequent determinant values:\n")
  freq_table <- sort(table(determinants_exact), decreasing = TRUE)
  print(head(freq_table, 10))
}
Top 10 most frequent determinant values:
determinants_exact
 -45   45  -27   27  -55   55  -65  -11   11   65 
1704 1704 1611 1611 1503 1503 1500 1500 1500 1500

Symmetry

Code 
cat(sprintf("Skewness: %.3f\n", moments::skewness(determinants_exact)))
Skewness: 0.000

Why this shape? An analytical perspective

The two most striking features of the distribution — perfect symmetry around zero and the tent-like (near-triangular) shape — both have mathematical explanations rooted in the structure of the determinant.

Why symmetric?

For any filling of the 3×3 matrix with a permutation of \(\{1, \ldots, 9\}\) that yields \(\det(\mathbf{A}) = d\), swapping any two rows produces another valid filling (still a permutation of \(\{1, \ldots, 9\}\)) with \(\det = -d\). This gives an exact bijection between fillings with determinant \(d\) and fillings with determinant \(-d\), so the distribution is perfectly symmetric around zero — not approximately, but exactly.

Why tent-shaped?

The Leibniz formula expands the 3×3 determinant into exactly six signed products, three positive and three negative:

\[ \det(\mathbf{A}) = \underbrace{(a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32})}_{P} - \underbrace{(a_{13}a_{22}a_{31} + a_{11}a_{23}a_{32} + a_{12}a_{21}a_{33})}_{N} \]

Each of \(P\) and \(N\) is a sum of three products, one from each row and each column — a “Latin square transversal” of the matrix. By the same even/odd permutation symmetry, \(P\) and \(N\) have identical marginal distributions, so \(\det = P - N\) is symmetric.

The shape of \(P - N\) depends on how close to uniform the marginal distribution of \(P\) (or \(N\)) is. If \(P\) were exactly uniform on some interval \([a, b]\), the difference of two independent copies would be exactly triangular (the Irwin–Hall \(n=2\) case). \(P\) and \(N\) aren’t independent — they share the same nine entries — but their marginal distributions are bounded, roughly unimodal, and symmetric enough that the difference is close to triangular rather than Gaussian.

The distribution never approaches Gaussian here because:

  • The support is bounded: determinant values are constrained to a finite range.
  • Only six terms contribute; the Central Limit Theorem requires many independent summands.
  • The terms are not independent: entries of \(\mathbf{A}\) are a permutation, so all nine values constrain each other.

An exact analytic formula?

Conclusions

This analysis reveals the distribution of determinants when all possible 3×3 matrices are formed from the digits 1-9, each used exactly once. The Monte Carlo approximation with random permutations closely matches the exact distribution, validating the sampling approach for similar problems where exact enumeration might be computationally prohibitive.