Rotate a Matrix by 90 Degrees Clockwise Optimal In-place Algorithm

Problem Statement❯

Given a square matrix of size n x n, rotate the entire matrix by 90 degrees in the clockwise direction. The rotation should be done in-place, meaning no extra matrix should be used for the transformation.

Each element of the matrix should be repositioned so that after rotation, the first row becomes the last column, the second row becomes the second-last column, and so on.
If the matrix is empty or has only one element, return it as is.

Examples❯

Input Matrix	Output Matrix	Description
[[1,2],[3,4]]	[[3,1],[4,2]]	2x2 matrix rotated 90° clockwise
[[1,2,3],[4,5,6],[7,8,9]]	[[7,4,1],[8,5,2],[9,6,3]]	Standard 3x3 matrix rotation
[[1]]	[[1]]	Single element matrix remains unchanged
[]	[]	Empty matrix has no elements to rotate
[[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]	[[13,9,5,1],[14,10,6,2],[15,11,7,3],[16,12,8,4]]	4x4 matrix rotated in-place

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Solution❯

To rotate a square matrix by 90 degrees clockwise without using extra space, we need to reposition the elements in-place. Instead of rotating each layer individually (which is another method), this optimal approach uses two simple steps:

Step 1: Transpose the Matrix

Transposing a matrix means converting rows into columns. We swap every element matrix[i][j] with matrix[j][i] for all i < j. After this step, the matrix is flipped along its diagonal.

Step 2: Reverse Each Row

Now, we reverse each row in the transposed matrix. This aligns all the elements to their new 90-degree rotated positions.

Why This Works

When we transpose the matrix, the top row becomes the left column. But to fully rotate it clockwise, we still need to move the last column to the top, which is achieved by reversing each row after transposition.

Let’s Discuss Different Scenarios

Normal Square Matrix: For matrices like 2x2, 3x3, or 4x4, the algorithm works seamlessly and efficiently, with no extra space used.
Matrix with One Element: Rotating a 1x1 matrix has no visible effect. The matrix stays the same.
Empty Matrix: There's nothing to rotate. The function should simply return an empty list or matrix.
Non-Square Matrix: This approach only works on square matrices (equal number of rows and columns). If the matrix is not square, this logic should not be applied as it would produce incorrect results.

Because we don’t use any extra space, this is an in-place O(1) space and O(n²) time solution, where n is the number of rows (or columns).

Algorithm Steps❯

Given an n x n square matrix.
Step 1: Transpose the matrix: swap matrix[i][j] with matrix[j][i] for all i < j.
Step 2: Reverse each row of the matrix.
The matrix is now rotated 90 degrees clockwise.

Code

Python

JavaScript

Java

C++

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

    # Reverse each row
    for row in matrix:
        row.reverse()

    return matrix

# Sample Input
mat = [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
]
rotated = rotate_matrix(mat)
for row in rotated:
    print(row)

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