To solve this problem, we need to generate an n x n permutation matrix—a square matrix where each row and column contains exactly one 1 (and the rest 0s). This is achieved by using a permutation of indices (0 to n-1) to place the 1s in the matrix.

Approach

1. Initialize the Matrix: Create an n x n matrix filled with 0s using list comprehensions.
2. Choose a Permutation: A permutation of indices (0 to n-1) ensures each row and column gets exactly one 1. For example, a permutation like [2,0,1] for n=3 means row 0 has a 1 at column 2, row 1 at column 0, and row 2 at column 1.
3. Fill the Matrix: For each row index i, set the element at column permutation[i] to 1.

Solution Code

n = 3  # Example size (can be any positive integer)
permutation = [2, 0, 1]  # Example permutation (valid for n=3)

# Initialize n x n matrix with 0s
matrix = [[0] * n for _ in range(n)]

# Place 1s according to the permutation
for i in range(n):
    matrix[i][permutation[i]] = 1

# Print the result
for row in matrix:
    print(row)

Explanation

• Matrix Initialization: [[0]*n for _ in range(n)] creates a 2D list where each sublist (row) has n zeros.
• Permutation: The permutation list defines where to place the 1 in each row. For a valid permutation matrix, the permutation must be a bijection (each index from 0 to n-1 appears exactly once).
• Filling the Matrix: The loop iterates over each row, setting the 1 at the column specified by the permutation for that row. This ensures each row and column has exactly one 1.

Example Output (for n=3 and permutation [2,0,1])

[0, 0, 1]
[1, 0, 0]
[0, 1, 0]

Random Permutation (Optional)

To generate a random permutation matrix, use the random module:

import random

n = 5
permutation = random.sample(range(n), n)  # Random permutation of 0..n-1

matrix = [[0]*n for _ in range(n)]
for i in range(n):
    matrix[i][permutation[i]] = 1

for row in matrix:
    print(row)

This code will produce a random permutation matrix where each row and column has exactly one 1.

The key takeaway is that a permutation matrix is defined by a permutation of indices, and the code correctly implements this logic. ✅

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