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, \(\text{det}(\mathbf{A}) \equiv | \mathbf{A} |\). …

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?

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)

# Function to compute determinant from a vector of 9 numbers
det_from_vec <- function(v) {
  M <- matrix(v, nrow = 3, ncol = 3, byrow = TRUE)
  det(M)
}

Exact Solution: All 9! Permutations

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

Code 
library(arrangements)

# If arrangements package is not available, use base R
if (!requireNamespace("arrangements", quietly = TRUE)) {
  if (requireNamespace("gtools", quietly = TRUE)) {
    all_perms <- gtools::permutations(9, 9, 1:9)
    determinants_exact <- apply(all_perms, 1, det_from_vec)
  } else {
    cat("For exact solution, install 'arrangements' or 'gtools' package\n")
    determinants_exact <- NULL
  }
} else {
  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 
if (!is.null(determinants_exact)) {
  if (requireNamespace("moments", quietly = TRUE)) {
    cat(sprintf("Skewness: %.3f\n", moments::skewness(determinants_exact)))
    cat("  (0 = symmetric, >0 = right-skewed, <0 = left-skewed)\n")
  }
}
Skewness: 0.000
  (0 = symmetric, >0 = right-skewed, <0 = left-skewed)

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.