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How it works

We want to rotate an n x n matrix 90 degrees clockwise in-place.

A very clean way to do this is to break the rotation into two simple operations:

1. Reverse the matrix vertically
2. Transpose the matrix

Why this works:

• Reversing the matrix flips it upside down
• Transposing swaps rows with columns
• Doing both together results in a 90° clockwise rotation

This method is elegant, easy to remember, and avoids extra space.

Steps

1. Reverse the matrix vertically:
   • the first row becomes the last
   • the last row becomes the first
2. Transpose the matrix:
   • swap elements across the main diagonal
   • for all i < j, swap matrix[i][j] with matrix[j][i]
3. The matrix is now rotated 90 degrees clockwise in-place.

Complexity

• Time complexity: O(n ^ 2)
• Space complexity: O(1)

Things to watch for

Confusing Clockwise and Counter-Clockwise Rotation

The order of operations matters: for 90-degree clockwise rotation, reverse rows first then transpose. For counter-clockwise, transpose first then reverse rows. Mixing up this order or using the wrong sequence produces incorrect rotations.

Transposing the Entire Matrix Instead of Upper Triangle

When transposing in-place, swapping all pairs (i, j) with (j, i) including when i > j swaps each element twice, returning to the original matrix. The inner loop should only iterate for j > i to swap each pair exactly once.

Incorrect Index Calculation in Layer-by-Layer Rotation

In the four-cell rotation approach, computing the indices for the four corners incorrectly causes elements to be placed in wrong positions. Each of the four positions involves different combinations of l, r, i, top, and bottom, and confusing these leads to corrupted matrix values.

from typing import List

class Solution:
    def rotate(self, matrix: List[List[int]]) -> None:
        # Reverse the matrix vertically
        matrix.reverse()

        # Transpose the matrix
        for i in range(len(matrix)):
            for j in range(i + 1, len(matrix)):
                matrix[i][j], matrix[j][i] = matrix[j][i], matrix[i][j]

class Solution {
public:
    void rotate(vector<vector<int>>& matrix) {
        // Reverse the matrix vertically
        reverse(matrix.begin(), matrix.end());

        // Transpose the matrix
        for (int i = 0; i < matrix.size(); ++i) {
            for (int j = i + 1; j < matrix[i].size(); ++j)
                swap(matrix[i][j], matrix[j][i]);
        }
    }
};