Rotate Matrix by 90° Anti-Clockwise

Pattern:

Idea:

Variations :

• part of Matrices Everywhere !!!
• transpose-of-a-matrix

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💻 Code

This is transposing a square matrix (but a general case do exist)

def rotate_anticlockwise(matrix):
    n = len(matrix)

    # Step 1: Transpose in-place
    for i in range(n):
        for j in range(i + 1, n):
            matrix[i][j], matrix[j][i] = (
                matrix[j][i],
                matrix[i][j]
            )

    # Step 2: Reverse every column
    for col in range(n):
        top = 0
        bottom = n - 1

        while top < bottom:
            matrix[top][col], matrix[bottom][col] = (
                matrix[bottom][col],
                matrix[top][col]
            )
            top += 1
            bottom -= 1

    return matrix

Time complexity - O($n^2$)

Aux. Space complexity - O(1)

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Rotate Matrix by 90° Anti-Clockwise

Tags: #Matrix #Array #2D-Array #InPlace #Transpose #Reverse #Matrix-Manipulation #LC48 #FAANG

Problem Statement

Given an n × n square matrix, rotate it 90° anti-clockwise (counter-clockwise) in-place.

Example

Input:

1 2 3
4 5 6
7 8 9

Output:

3 6 9
2 5 8
1 4 7

Constraint: The matrix must be modified in-place using O(1) auxiliary space.

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Key Idea

A 90° anti-clockwise rotation can be decomposed into two simple matrix operations:

Rotate 90° Anti-Clockwise=Transpose+Reverse Every Column\text{Rotate 90° Anti-Clockwise} = \text{Transpose} + \text{Reverse Every Column}

This is the exact counterpart of clockwise rotation:

Rotation	Operations
90° Clockwise	Transpose → Reverse each row
90° Anti-clockwise	Transpose → Reverse each column

This relationship is one of the most important matrix interview patterns.

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Why Does This Work?

Consider the original matrix:

1 2 3
4 5 6
7 8 9

Step 1 — Transpose

Swap across the main diagonal:

1 4 7
2 5 8
3 6 9

Rows became columns.

Step 2 — Reverse Every Column

Reverse each vertical column:

3 6 9
2 5 8
1 4 7

Exactly the required anti-clockwise rotation.

Visual intuition

Original
1 2 3
4 5 6
7 8 9

      │
      ▼  Transpose

1 4 7
2 5 8
3 6 9

      │
      ▼  Reverse Columns

3 6 9
2 5 8
1 4 7

The transpose changes the orientation, and reversing columns completes the rotation.

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Approach 1 — In-Place (Transpose + Reverse Columns)

Algorithm

1. Transpose the square matrix.

2. Reverse every column.

3. Return the modified matrix.

Python Solution

def rotate_anticlockwise(matrix):
    n = len(matrix)

    # Step 1: Transpose in-place
    for i in range(n):
        for j in range(i + 1, n):
            matrix[i][j], matrix[j][i] = (
                matrix[j][i],
                matrix[i][j]
            )

    # Step 2: Reverse every column
    for col in range(n):
        top = 0
        bottom = n - 1

        while top < bottom:
            matrix[top][col], matrix[bottom][col] = (
                matrix[bottom][col],
                matrix[top][col]
            )
            top += 1
            bottom -= 1

    return matrix

Dry Run

Initial matrix:

1 2 3
4 5 6
7 8 9

After transpose:

1 4 7
2 5 8
3 6 9

Reverse Column 0

1
2
3

↓

3
2
1

Matrix becomes:

3 4 7
2 5 8
1 6 9

Reverse Column 1

4
5
6

↓

6
5
4

Matrix:

3 6 7
2 5 8
1 4 9

Reverse Column 2

7
8
9

↓

9
8
7

Final:

3 6 9
2 5 8
1 4 7

Complexity

Time Complexity

• Transpose: $O(n^2)$

• Reverse columns: $O(n^2)$

Overall:

O(n2)O(n^2)

Auxiliary Space

O(1)O(1)

Output Space

None — the input matrix is modified in-place.

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Approach 2 — Create a New Matrix

If in-place modification is not required, directly place every element into its rotated position.

Position Mapping

For anti-clockwise rotation:

(i,j)→(n−1−j, i)(i, j) \rightarrow (n-1-j,\ i)

Example:

matrix[0][2] = 3

goes to

result[0][0] = 3

Python Solution

def rotate_anticlockwise(matrix):
    n = len(matrix)

    result = [[0] * n for _ in range(n)]

    for i in range(n):
        for j in range(n):
            result[n - 1 - j][i] = matrix[i][j]

    return result

Why the Formula Works

Take the element 9:

Position = (2,2)

Using:

(n−1−j, i)(n-1-j,\ i)

we get:

(3-1-2, 2)
= (0,2)

which is exactly where 9 appears after anti-clockwise rotation.

Complexity

Time Complexity

O(n2)O(n^2)

Auxiliary Space

The new matrix stores every element:

O(n2)O(n^2)

Output Space

O(n2)O(n^2)

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Important Formulae

Clockwise Rotation

$$(i,j)\rightarrow(j,n-1-i)$$

Equivalent operations:

$\text{Transpose} + \text{Reverse Rows}$

Anti-Clockwise Rotation

$(i,j)\rightarrow(n-1-j,i)$

Equivalent operations:

$\text{Transpose} + \text{Reverse Columns}$

Memory Shortcut

Clockwise
Transpose
↓
Reverse Rows

Anti-clockwise
Transpose
↓
Reverse Columns

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Common Mistakes

Mistake 1 — Reversing rows instead of columns

After transpose:

1 4 7
2 5 8
3 6 9

If you reverse rows:

7 4 1
8 5 2
9 6 3

This is clockwise, not anti-clockwise.

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Mistake 2 — Transposing the entire matrix twice

Incorrect:

for i in range(n):
    for j in range(n):
        swap(...)

This swaps every pair twice.

Correct:

for i in range(n):
    for j in range(i + 1, n):

Only traverse the upper triangle.

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Mistake 3 — Applying to a rectangular matrix

The in-place algorithm only works for square matrices.

A 2 × 3 matrix becomes 3 × 2, so dimensions change and a new matrix is required.

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Pattern Recognition

The Matrix Transformation Family

Problem	Formula	Operations
Transpose	$(i,j)\rightarrow(j,i)$	Swap across diagonal
Rotate 90° CW	$(i,j)\rightarrow(j,n-1-i)$	Transpose + Reverse Rows
Rotate 90° CCW	$(i,j)\rightarrow(n-1-j,i)$	Transpose + Reverse Columns

Instead of memorizing all three independently, remember transpose as the foundational operation.

Interview Takeaway

Whenever you hear:

Rotate a square matrix by 90°

Immediately think:

1. Is it clockwise or anti-clockwise?

2. Perform an in-place transpose.

3. Reverse rows (CW) or columns (CCW).

One-line memory hook: Anti-clockwise = Transpose → Reverse Columns.