Rotate Image

Rotate a square matrix 90 degrees clockwise in place.

Start with the concrete trace below. It shows the state the algorithm must carry as it runs.

Problem trace

Rotate Image: Reverse row order, then swap every above-diagonal cell with its reflected below-diagonal partner.

Input and goalRotate a square matrix 90 degrees clockwise in place.

Start with the original coordinatesThe 3x3 input is [[1,2,3],[4,5,6],[7,8,9]] before any in-place mutation.

1value 12value 23value 34value 45value 56value 67value 78value 89value 9

actioninitialize

Recognize it

Use this transformation for a square matrix that must rotate 90 degrees clockwise without allocating a second matrix.

Keep true

After row reversal, values have moved from (r,c) to (n-1-r,c); after transposition, each reaches (c,n-1-r), exactly its clockwise coordinate.

Reuse it

Derive a target coordinate mapping, then factor it into simple reversible transforms whose in-place loops visit each swap pair exactly once.

Pattern: Reverse rows, then transpose.

Simple idea: Reversing row order moves the bottom to the top. Swapping across the main diagonal then puts every value in its clockwise position.

def rotate_image(matrix: list[list[int]]) -> None:
   matrix.reverse()
   for row in range(len(matrix)):
      for col in range(row + 1, len(matrix)):
         matrix[row][col], matrix[col][row] = matrix[col][row], matrix[row][col]

Cost: $O(n^2)$ time and $O(1)$ space.