Complete Guide On 2D Array (Matrix) Rotations – Data Structure and Algorithms Tutorial

Matrix Rotation is defined as rotating the elements of the array around its center in a clockwise or counterclockwise direction, or rotating the elements by some position in clockwise or anticlockwise or interchanging of rows and columns. 

There are several ways to perform rotations on a Matrix, including using loops, using built-in functions, etc.

What is 2D Array (Matrix) Rotation Types of Matrix Rotation: 1. Rotate the Matrix (90 degrees clockwise) 1: Rotate by Cycles 2: Rotate through diagonal and middle 2. Rotate Matrix elements by one place 3. Rotations in Matrix by row or column ( Transpose of the matrix ) 1. Using Extra Space 2. Using Constant Space Applications of Matrix Rotation Some FAQs regarding Matrix Rotation Advantages of Rotations in a Matrix Some Popular Interview Problems on Matrix Rotation Some Important Problems/Article links of Matrix Rotation

Types of Matrix Rotation:

The rotation of a Matrix can be classified into the following categories:

1. Rotate the matrix 90 degrees (or multiples of 90) clockwise or counterclockwise.
2. Rotate matrix elements by 1 (or more) places to the right or left.
3. Interchange rows and columns

The details about each type are discussed below with some examples for better understanding.

1. Rotate the Matrix (90 degrees clockwise):

Rotate the Matrix by 90 degrees clockwise

1: Rotate by Cycles

Intuition:

To solve the question without any extra space, rotate the array in form of squares, dividing the matrix into squares or cycles. 

For example, A 4 X 4 matrix will have 2 cycles. The first cycle is formed by its 1st row, last column, last row, and 1st column. The second cycle is formed by the 2nd row, second-last column, second-last row, and 2nd column.

The idea is for each square cycle, to swap the elements involved with the corresponding cell in the matrix in an clockwise direction i.e. from top to right, right to bottom, bottom to left, and from left to top one at a time using nothing but a temporary variable to achieve this

Illustration:

Let size of row and column be 3. 
During first iteration –
a[i][j] = Element at first index (leftmost corner top)= 1.
a[j][n-1-i]= Rightmost corner top Element = 3.
a[n-1-i][n-1-j] = Rightmost corner bottom element = 9.
a[n-1-j][i] = Leftmost corner bottom element = 7.
Move these elements in the clockwise direction.

During second iteration –
a[i][j] = 2.
a[j][n-1-i] = 6.
a[n-1-i][n-1-j] = 8.
a[n-1-j][i] = 4. 
Similarly, move these elements in the clockwise direction. 

Below is the implementation of the above approach:

C++

// C++ implementation of above approach
#include <bits/stdc++.h>
using namespace std;

#define N 3

// Function to rotate the matrix 90 degree clockwise
void rotate90Clockwise(int a[N][N])
{

    // Traverse each cycle
    for (int i = 0; i < N / 2; i++) {
        for (int j = i; j < N - i - 1; j++) {

            // Swap elements of each cycle
            // in clockwise direction
            int temp = a[i][j];
            a[i][j] = a[N - 1 - j][i];
            a[N - 1 - j][i] = a[N - 1 - i][N - 1 - j];
            a[N - 1 - i][N - 1 - j] = a[j][N - 1 - i];
            a[j][N - 1 - i] = temp;
        }
    }
}

// Function for print matrix
void printMatrix(int arr[N][N])
{
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++)
            cout << arr[i][j] << " ";
        cout << '\n';
    }
}

// Driver code
int main()
{
    int arr[N][N]
        = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };
    rotate90Clockwise(arr);
    printMatrix(arr);
    return 0;
}

Java

// Java implementation of above approach
import java.io.*;

class GFG {

    static int N = 3;

    // Function to rotate the matrix 90 degree clockwise
    static void rotate90Clockwise(int a[][])
    {

        // Traverse each cycle
        for (int i = 0; i < N / 2; i++) {
            for (int j = i; j < N - i - 1; j++) {

                // Swap elements of each cycle
                // in clockwise direction
                int temp = a[i][j];
                a[i][j] = a[N - 1 - j][i];
                a[N - 1 - j][i] = a[N - 1 - i][N - 1 - j];
                a[N - 1 - i][N - 1 - j] = a[j][N - 1 - i];
                a[j][N - 1 - i] = temp;
            }
        }
    }

    // Function for print matrix
    static void printMatrix(int arr[][])
    {
        for (int i = 0; i < N; i++) {
            for (int j = 0; j < N; j++)
                System.out.print(arr[i][j] + " ");
            System.out.println();
        }
    }

    // Driver code

    public static void main(String[] args)
    {
        int arr[][]
            = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };
        rotate90Clockwise(arr);
        printMatrix(arr);
    }
}

Python3

# Python code to implement the above approach


# Function to rotate matrix 90 degree clockwise
def rotate90Clockwise(A):
    N = len(A[0])
    for i in range(N // 2):
        for j in range(i, N - i - 1):
            temp = A[i][j]
            A[i][j] = A[N - 1 - j][i]
            A[N - 1 - j][i] = A[N - 1 - i][N - 1 - j]
            A[N - 1 - i][N - 1 - j] = A[j][N - 1 - i]
            A[j][N - 1 - i] = temp


# Function to print the matrix
def printMatrix(A):
    N = len(A[0])
    for i in range(N):
        print(A[i])


# Driver code
if __name__ == '__main__':
    A = [[1, 2, 3],
         [4, 5, 6],
         [7, 8, 9]]
    rotate90Clockwise(A)
    printMatrix(A)

C#

// C# implementation of above approach

using System;

class GFG {
    static int N = 3;

    // Function to rotate the matrix 90 degree clockwise
    static void Rotate90Clockwise(int[, ] a)
    {

        // Traverse each cycle
        for (int i = 0; i < N / 2; i++) {
            for (int j = i; j < N - i - 1; j++) {

                // Swap elements of each cycle
                // in clockwise direction
                int temp = a[i, j];
                a[i, j] = a[N - 1 - j, i];
                a[N - 1 - j, i] = a[N - 1 - i, N - 1 - j];
                a[N - 1 - i, N - 1 - j] = a[j, N - 1 - i];
                a[j, N - 1 - i] = temp;
            }
        }
    }

    // Function for print matrix
    static void PrintMatrix(int[, ] arr)
    {
        for (int i = 0; i < N; i++) {
            for (int j = 0; j < N; j++)
                Console.Write(arr[i, j] + " ");
            Console.WriteLine();
        }
    }

    // Driver code
    static void Main(string[] args)
    {
        int[, ] arr
            = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };
        Rotate90Clockwise(arr);
        PrintMatrix(arr);
    }
}

Javascript

const N = 3;

// Function to rotate the matrix 90 degree clockwise
function rotate90Clockwise(a) {

    // Traverse each cycle
    for (let i = 0; i < N / 2; i++) {
        for (let j = i; j < N - i - 1; j++) {

            // Swap elements of each cycle
            // in clockwise direction
            let temp = a[i][j];
            a[i][j] = a[N - 1 - j][i];
            a[N - 1 - j][i] = a[N - 1 - i][N - 1 - j];
            a[N - 1 - i][N - 1 - j] = a[j][N - 1 - i];
            a[j][N - 1 - i] = temp;
        }
    }
}

// Function for print matrix
function printMatrix(arr) {
    for (let i = 0; i < N; i++) {
        let row = "";
        for (let j = 0; j < N; j++) {
            row += arr[i][j] + " ";
        }
        console.log(row);
    }
}

// Driver code
let arr = [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ];
rotate90Clockwise(arr);
printMatrix(arr);

Output

7 4 1 
8 5 2 
9 6 3 

Time Complexity: O(n*n)
Auxiliary Space: O(1)

2: Rotate through diagonal and middle:

The Approach is to rotate the given matrix two times, first time with respect to the Main diagonal, next time rotate the resultant matrix with respect to the middle column, Consider the following illustration to have a clear insight into it.

Illustration:

Consider the following matrix.

Original Matrix

• Swap all elements which are on opposite sides of the main diagonal

Rotate through the diagonal

• Next, swap elements which are on either sides of the middle column, and we get the desired matrix.

Rotate through the middle

Below is the implementation of the above approach:

C++

// C++ code to implement above approach
#include <bits/stdc++.h>
using namespace std;

#define N 3

void print(int arr[N][N])
{
    for (int i = 0; i < N; ++i) {
        for (int j = 0; j < N; ++j)
            cout << arr[i][j] << " ";

        cout << '\n';
    }
}

void rotate(int arr[N][N])
{

    // First rotation
    // with respect to main diagonal
    for (int i = 0; i < N; ++i) {
        for (int j = 0; j < i; ++j) {
            int temp = arr[i][j];
            arr[i][j] = arr[j][i];
            arr[j][i] = temp;
        }
    }

    // Second rotation
    // with respect to middle column
    for (int i = 0; i < N; ++i) {
        for (int j = 0; j < N / 2; ++j) {
            int temp = arr[i][j];
            arr[i][j] = arr[i][N - j - 1];
            arr[i][N - j - 1] = temp;
        }
    }
}

// Driver code
int main()
{
    int arr[N][N]
        = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };
    rotate(arr);
    print(arr);
    return 0;
}

Java

// Java code to implement above approach
import java.io.*;

class GFG {

    static void rotate(int[][] arr)
    {

        int n = arr.length;

        // first rotation
        // with respect to main diagonal
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < i; ++j) {
                int temp = arr[i][j];
                arr[i][j] = arr[j][i];
                arr[j][i] = temp;
            }
        }
        // Second rotation
        // with respect to middle column
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < n / 2; ++j) {
                int temp = arr[i][j];
                arr[i][j] = arr[i][n - j - 1];
                arr[i][n - j - 1] = temp;
            }
        }
    }

    // to print matrix
    static void printMatrix(int arr[][])
    {
        int n = arr.length;
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++)
                System.out.print(arr[i][j] + " ");
            System.out.println();
        }
    }
    // Driver code
    public static void main(String[] args)
    {
        int arr[][]
            = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };
        rotate(arr);
        printMatrix(arr);
    }
}

Python3

# Python3 implementation of above approach
N = 3


# Function to rotate the matrix 90 degree clockwise
def rotate(arr):
    global N

    # First rotation
    # with respect to main diagonal
    for i in range(N):
        for j in range(i):
            temp = arr[i][j]
            arr[i][j] = arr[j][i]
            arr[j][i] = temp

    # Second rotation
    # with respect to middle column
    for i in range(N):
        for j in range(int(N/2)):
            temp = arr[i][j]
            arr[i][j] = arr[i][N-j-1]
            arr[i][N-j-1] = temp


# Driver code
if __name__ == '__main__':
    arr = [[1, 2, 3],
           [4, 5, 6],
           [7, 8, 9]]

    rotate(arr)

    for i in range(N):
        for j in range(N):
            print(arr[i][j], end=" ")
        print()

C#

// C# code to implement above approach

using System;

class GFG {
    static int N = 3;

    // Driver's code
    static void Main()
    {
          // Input array
        int[,] arr = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };

          // Function call
        Rotate(arr);
        Print(arr);
    }

      // Function to rotate the matrix 90 degree clockwise
    static void Rotate(int[,] arr)
    {
        // First rotation with respect to main diagonal
        for (int i = 0; i < N; ++i)
        {
            for (int j = 0; j < i; ++j)
            {
                int temp = arr[i,j];
                arr[i,j] = arr[j,i];
                arr[j,i] = temp;
            }
        }

        // Second rotation with respect to middle column
        for (int i = 0; i < N; ++i)
        {
            for (int j = 0; j < N / 2; ++j)
            {
                int temp = arr[i,j];
                arr[i,j] = arr[i,N - j - 1];
                arr[i,N - j - 1] = temp;
            }
        }
    }

      // Function to print the final array
    static void Print(int[,] arr)
    {
        for (int i = 0; i < N; ++i)
        {
            for (int j = 0; j < N; ++j)
                Console.Write(arr[i,j] + " ");

            Console.WriteLine();
        }
    }
}

Javascript

// JavaScript code to implement above approach

const N = 3;

// function to print the matrix
function print(arr)
{
    for (let i = 0; i < N; i++) {
        let row = "";
        for (let j = 0; j < N; j++) {
            row += arr[i][j] + " ";
        }
        console.log(row);
    }
}

// function to rotate the matrix
function rotate(arr)
{
    // First rotation with respect to main diagonal
    for (let i = 0; i < N; ++i) {
        for (let j = 0; j < i; ++j) {
            let temp = arr[i][j];
            arr[i][j] = arr[j][i];
            arr[j][i] = temp;
        }
    }

    // Second rotation with respect to middle column
    for (let i = 0; i < N; ++i) {
        for (let j = 0; j < N / 2; ++j) {
            let temp = arr[i][j];
            arr[i][j] = arr[i][N - j - 1];
            arr[i][N - j - 1] = temp;
        }
    }
}

// Driver code
let arr = [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ];
rotate(arr);
print(arr);

Output

7 4 1 
8 5 2 
9 6 3 

Time Complexity: O(n * n)
Auxiliary Space: O(1)

2. Rotate Matrix elements by one place:

• We have the original matrix.

Rotate matrix elements by one place

• Rotate by one place to the right

Rotate matrix elements by one place

• Final matrix after the shifts.

Rotate matrix elements by one place

Intuition:

The idea is to use loops similar to the program for printing a matrix in spiral form. 
One by one rotate all rings of elements, starting from the outermost. To rotate a ring, we need to do the following.

• Move elements of top row. 
• Move elements of last column. 
• Move elements of bottom row.
• Move elements of first column.
• Repeat above steps for inner ring while there is an inner ring.

Below is the implementation of the above approach:

C++

// C++ program to implement the above approach

#include <bits/stdc++.h>
#define R 3
#define C 3
using namespace std;

// A function to rotate a matrix mat[][] of size R x C.
// Initially, m = R and n = C
void rotateMatrix(int m, int n, int mat[R][C])
{
    int row = 0, col = 0;
    int prev, curr;

    while (row < m && col < n) {

        if (row + 1 == m || col + 1 == n)
            break;

        // Store the first element of next row, this
        // element will replace first element of current
        // row
        prev = mat[row + 1][col];

        // Move elements of first row
        // from the remaining rows
        for (int i = col; i < n; i++) {
            curr = mat[row][i];
            mat[row][i] = prev;
            prev = curr;
        }
        row++;

        // Move elements of last column
        // from the remaining columns
        for (int i = row; i < m; i++) {
            curr = mat[i][n - 1];
            mat[i][n - 1] = prev;
            prev = curr;
        }
        n--;

        // Move elements of last row
        // from the remaining rows
        if (row < m) {
            for (int i = n - 1; i >= col; i--) {
                curr = mat[m - 1][i];
                mat[m - 1][i] = prev;
                prev = curr;
            }
        }
        m--;

        // Move elements of first column
        // from the remaining rows
        if (col < n) {
            for (int i = m - 1; i >= row; i--) {
                curr = mat[i][col];
                mat[i][col] = prev;
                prev = curr;
            }
        }
        col++;
    }

    // Print rotated matrix
    for (int i = 0; i < R; i++) {
        for (int j = 0; j < C; j++)
            cout << mat[i][j] << " ";
        cout << endl;
    }
}

// Driver code
int main()
{
    // Input Matrix
    int a[R][C] = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };

    rotateMatrix(R, C, a);
    return 0;
}

Java

// Java program to implement above approach
import java.lang.*;
import java.util.*;

class GFG {
    static int R = 3;
    static int C = 3;

    // A function to rotate a matrix
    // mat[][] of size R x C.
    // Initially, m = R and n = C
    static void rotateMatrix(int m, int n, int mat[][])
    {
        int row = 0, col = 0;
        int prev, curr;

        /*
        row - Starting row index
        m - ending row index
        col - starting column index
        n - ending column index
        i - iterator
        */
        while (row < m && col < n) {

            if (row + 1 == m || col + 1 == n)
                break;

            // Store the first element of next
            // row, this element will replace
            // first element of current row
            prev = mat[row + 1][col];

            // Move elements of first row
            // from the remaining rows
            for (int i = col; i < n; i++) {
                curr = mat[row][i];
                mat[row][i] = prev;
                prev = curr;
            }
            row++;

            // Move elements of last column
            // from the remaining columns
            for (int i = row; i < m; i++) {
                curr = mat[i][n - 1];
                mat[i][n - 1] = prev;
                prev = curr;
            }
            n--;

            // Move elements of last row
            // from the remaining rows
            if (row < m) {
                for (int i = n - 1; i >= col; i--) {
                    curr = mat[m - 1][i];
                    mat[m - 1][i] = prev;
                    prev = curr;
                }
            }
            m--;

            // Move elements of first column
            // from the remaining rows
            if (col < n) {
                for (int i = m - 1; i >= row; i--) {
                    curr = mat[i][col];
                    mat[i][col] = prev;
                    prev = curr;
                }
            }
            col++;
        }

        // Print rotated matrix
        for (int i = 0; i < R; i++) {
            for (int j = 0; j < C; j++)
                System.out.print(mat[i][j] + " ");
            System.out.print("\n");
        }
    }

    /* Driver program to test above functions */
    public static void main(String[] args)
    {
        // input matrix
        int a[][]
            = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };

        rotateMatrix(R, C, a);
    }
}

Python3

# Python program to implement above approach


# Function to rotate a matrix
def rotateMatrix(mat):

    if not len(mat):
        return
    top = 0
    bottom = len(mat)-1

    left = 0
    right = len(mat[0])-1

    while left < right and top < bottom:

        # Store the first element of next row,
        # this element will replace first element of
        # current row
        prev = mat[top+1][left]

        # Move elements of top row one step right
        for i in range(left, right+1):
            curr = mat[top][i]
            mat[top][i] = prev
            prev = curr

        top += 1

        # Move elements of rightmost column one step downwards
        for i in range(top, bottom+1):
            curr = mat[i][right]
            mat[i][right] = prev
            prev = curr

        right -= 1

        # Move elements of bottom row one step left
        for i in range(right, left-1, -1):
            curr = mat[bottom][i]
            mat[bottom][i] = prev
            prev = curr

        bottom -= 1

        # Move elements of leftmost column one step upwards
        for i in range(bottom, top-1, -1):
            curr = mat[i][left]
            mat[i][left] = prev
            prev = curr

        left += 1

    return mat


# Function to print matrix
def printMatrix(mat):
    for row in mat:
        print(row)


# Driver code
if __name__ == '__main__':
    matrix = [[1, 2, 3],
              [4, 5, 6],
              [7, 8, 9]]
    matrix = rotateMatrix(matrix)

    # Print modified matrix
    printMatrix(matrix)

C#

// C# program to implement the above approach

using System;

class GFG {
    static int R = 3;
    static int C = 3;

    // A function to rotate a matrix mat[][] of size R x C.
    // Initially, m = R and n = C
    static void RotateMatrix(int[,] mat, int m, int n) {
        int row = 0, col = 0;
        int prev, curr;

        while (row < m && col < n) {
            if (row + 1 == m || col + 1 == n)
                break;

            // Store the first element of next row, this
            // element will replace first element of current
            // row
            prev = mat[row + 1, col];

            // Move elements of first row
            // from the remaining rows
            for (int i = col; i < n; i++) {
                curr = mat[row, i];
                mat[row, i] = prev;
                prev = curr;
            }

            row++;

            // Move elements of last column
            // from the remaining columns
            for (int i = row; i < m; i++) {
                curr = mat[i, n - 1];
                mat[i, n - 1] = prev;
                prev = curr;
            }

            n--;

            // Move elements of last row
            // from the remaining rows
            if (row < m) {
                for (int i = n - 1; i >= col; i--) {
                    curr = mat[m - 1, i];
                    mat[m - 1, i] = prev;
                    prev = curr;
                }
            }

            m--;

              // Move elements of first column
            // from the remaining rows
            if (col < n) {
                for (int i = m - 1; i >= row; i--) {
                    curr = mat[i, col];
                    mat[i, col] = prev;
                    prev = curr;
                }
            }

            col++;
        }

        // Print rotated matrix
        for (int i = 0; i < R; i++) {
            for (int j = 0; j < C; j++) {
                Console.Write(mat[i, j] + " ");
            }

            Console.WriteLine();
        }
    }

      // Driver's code
    static void Main(string[] args) {
        // Input Matrix
        int[,] mat = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };

        RotateMatrix(mat, R, C);
    }
}

Javascript

const R = 3;
const C = 3;

// A function to rotate a matrix mat[][] of size R x C.
// Initially, m = R and n = C
function rotateMatrix(m, n, mat) {
    let row = 0, col = 0;
    let prev, curr;

    // Store the first element of next row, this
    // element will replace first element of current
    // row
    while (row < m && col < n) {
        if (row + 1 === m || col + 1 === n)
            break;

        prev = mat[row + 1][col];

        // Move elements of first row
        // from the remaining rows
        for (let i = col; i < n; i++) {
            curr = mat[row][i];
            mat[row][i] = prev;
            prev = curr;
        }
        row++;

        // Move elements of last column
        // from the remaining columns
        for (let i = row; i < m; i++) {
            curr = mat[i][n - 1];
            mat[i][n - 1] = prev;
            prev = curr;
        }
        n--;

        // Move elements of last row
        // from the remaining rows
        if (row < m) {
            for (let i = n - 1; i >= col; i--) {
                curr = mat[m - 1][i];
                mat[m - 1][i] = prev;
                prev = curr;
            }
        }
        m--;


        // Move elements of first column
        // from the remaining rows
        if (col < n) {
            for (let i = m - 1; i >= row; i--) {
                curr = mat[i][col];
                mat[i][col] = prev;
                prev = curr;
            }
        }
        col++;
    }
}

// Drive code
const mat = [
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]
];

rotateMatrix(R, C, mat);
console.log(mat);

Output

4 1 2 
7 5 3 
8 9 6 

Time Complexity: O(m*n) where m is the number of rows & n is the number of columns.
Auxiliary Space: O(1). 

3. Rotations in Matrix by row or column ( Transpose of the matrix ):

Transpose of a matrix is obtained by changing rows to columns and columns to rows. In other words, the transpose of A[N][M] is obtained by changing A[i][j] to A[j][i].

Transpose of the matrix

There are two methods to solve this kind of problem:

1. Using Extra Space

Intuition:

Run a nested loop using two integer pointers i and j for 0 <= i < N and 0 <= j < M
Set B[i][j] equal to A[j][i]

Below is the implementation of the above approach:

C++

// C++ program to implement the above approach
#include <bits/stdc++.h>
using namespace std;
#define N 3

// This function stores transpose
// of A[][] in B[][]
void transpose(int A[][N], int B[][N])
{
    int i, j;
    for (i = 0; i < N; i++)
        for (j = 0; j < N; j++)
            B[i][j] = A[j][i];
}

// Driver code
int main()
{
    int A[N][N] = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };

    // Note dimensions of B[][]
    int B[N][N], i, j;

    // Function call
    transpose(A, B);

    cout << "Modified matrix is \n";
    for (i = 0; i < N; i++) {
        for (j = 0; j < N; j++)
            cout << " " << B[i][j];

        cout << "\n";
    }
    return 0;
}

Java

// Java Program to implement the above approach

class GFG {
    static final int N = 3;

    // This function stores transpose
    // of A[][] in B[][]
    static void transpose(int A[][], int B[][])
    {
        int i, j;
        for (i = 0; i < N; i++)
            for (j = 0; j < N; j++)
                B[i][j] = A[j][i];
    }

    // Driver code
    public static void main(String[] args)
    {
        int A[][]
            = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };

        int B[][] = new int[N][N], i, j;

        // Function call
        transpose(A, B);

        System.out.print("Modified matrix is \n");
        for (i = 0; i < N; i++) {
            for (j = 0; j < N; j++)
                System.out.print(B[i][j] + " ");
            System.out.print("\n");
        }
    }
}

Python3

# Python3 code to implement the above approach

N = 3


# This function stores
# transpose of A[][] in B[][]
def transpose(A, B):
    for i in range(N):
        for j in range(N):
            B[i][j] = A[j][i]


# Driver code
if __name__ == '__main__':
    A = [[1, 2, 3],
         [4, 5, 6],
         [7, 8, 9]]

    # To store result
    B = [[0 for x in range(N)] for y in range(N)]

    # Function call
    transpose(A, B)

    print("Modified matrix is")
    for i in range(N):
        for j in range(N):
            print(B[i][j], " ", end='')
        print()

C#

using System;

class Program
{
    const int N = 3;

    // This function stores transpose
    // of A[][] in B[][]
    static void Transpose(int[,] A, int[,] B)
    {
        int i, j;
        for (i = 0; i < N; i++)
        {
            for (j = 0; j < N; j++)
            {
                B[i, j] = A[j, i];
            }
        }
    }

    static void Main()
    {
        int[,] A = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };

        // Note dimensions of B[,]
        int[,] B = new int[N, N];
        int i, j;

        // Function call
        Transpose(A, B);

        Console.WriteLine("Modified matrix is:");
        for (i = 0; i < N; i++)
        {
            for (j = 0; j < N; j++)
            {
                Console.Write(" " + B[i, j]);
            }
            Console.WriteLine();
        }
    }
}

Javascript

// Function to transpose a square matrix
function transpose(A) {
  // Get the size of the square matrix
  const N = A.length;
  // Create an empty array to store the transposed matrix
  const B = [];

  // Loop through the rows of the original matrix
  for (let i = 0; i < N; i++) {
    // Create a new row in the transposed matrix
    B[i] = [];
    // Loop through the columns of the original matrix
    for (let j = 0; j < N; j++) {
      // Swap the elements of the original matrix to create the transposed matrix
      // Rows become columns and columns become rows
      B[i][j] = A[j][i];
    }
  }

  // Return the transposed matrix
  return B;
}

// Driver code

// Define a 3x3 matrix 'A'
const A = [
  [1, 2, 3],
  [4, 5, 6],
  [7, 8, 9]
];

// Call the 'transpose' function to get the transposed matrix 'B'
const B = transpose(A);

// Display the transposed matrix 'B'
console.log("Modified matrix is:");
for (let i = 0; i < B.length; i++) {
  let rowStr = "";
  for (let j = 0; j < B[i].length; j++) {
    // Create a string representation of the matrix elements for printing
    rowStr += " " + B[i][j];
  }
  // Log the row of the transposed matrix
  console.log(rowStr);
}

Output

Modified matrix is 
 1 4 7
 2 5 8
 3 6 9

Time complexity: O(M x N).
Auxiliary Space: O(M x N), since M x N extra space has been used.

2. Using Constant Space:

Run a nested loop using two integer pointers i and j for 0 <= i < N and 0 <= j < M.
Swap A[i][j] with A[j][i].

Below is the implementation of the above approach:

C++

// C++ code to implement the above approach
#include <bits/stdc++.h>
using namespace std;

#define N 3

// Converts A[][] to its transpose
void transpose(int A[][N])
{
    for (int i = 0; i < N; i++)
        for (int j = i + 1; j < N; j++)
            swap(A[i][j], A[j][i]);
}

int main()
{
    int A[N][N] = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };

    transpose(A);

    printf("Modified matrix is \n");
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++)
            printf("%d ", A[i][j]);
        printf("\n");
    }

    return 0;
}

Java

// Java Program to implement the above approach

class GFG {
    static final int N = 3;

    // Finds transpose of A[][] in-place
    static void transpose(int A[][])
    {
        for (int i = 0; i < N; i++)
            for (int j = i + 1; j < N; j++) {
                int temp = A[i][j];
                A[i][j] = A[j][i];
                A[j][i] = temp;
            }
    }

    // Driver code
    public static void main(String[] args)
    {
        int A[][]
            = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };

        transpose(A);

        System.out.print("Modified matrix is \n");
        for (int i = 0; i < N; i++) {
            for (int j = 0; j < N; j++)
                System.out.print(A[i][j] + " ");
            System.out.print("\n");
        }
    }
}

Python3

# Python3 Program to implement the above approach

N = 3


# Finds transpose of A[][] in-place
def transpose(A):
    for i in range(N):
        for j in range(i+1, N):
            A[i][j], A[j][i] = A[j][i], A[i][j]


# Driver code
if __name__ == '__main__':
    A = [[1, 2, 3],
         [4, 5, 6],
         [7, 8, 9]]

    transpose(A)
    print("Modified matrix is")
    for i in range(N):
        for j in range(N):
            print(A[i][j],end=' ')
        print()

C#

// C# code to implement the above approach
using System;

class Program
{

      // Converts A[][] to its transpose
    static void Transpose(int[,] A)
    {
        int N = A.GetLength(0);
        for (int i = 0; i < N; i++)
        {
            for (int j = i + 1; j < N; j++)
            {
                int temp = A[i, j];
                A[i, j] = A[j, i];
                A[j, i] = temp;
            }
        }
    }

    static void Main()
    {
        int N = 3;
        int[,] A = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };
        Transpose(A);
        Console.WriteLine("Modified matrix is");
        for (int i = 0; i < N; i++)
        {
            for (int j = 0; j < N; j++)
            {
                Console.Write(A[i, j] + " ");
            }
            Console.WriteLine();
        }
    }
}

Javascript

// JavaScript code to implement matrix transposition

const N = 3;

// Function to transpose the matrix
function transpose(A) {
    for (let i = 0; i < N; i++) {
        for (let j = i + 1; j < N; j++) {
            // Swap A[i][j] and A[j][i]
            [A[i][j], A[j][i]] = [A[j][i], A[i][j]];
        }
    }
}

// Main function
function main() {
    let A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]];

    // Transpose the matrix
    transpose(A);

    console.log("Modified matrix is");
    for (let i = 0; i < N; i++) {
        for (let j = 0; j < N; j++) {
            console.log(A[i][j] + " ");
        }
        console.log();
    }
}

// Call the main function
main();

Output

Modified matrix is 
1 4 7 
2 5 8 
3 6 9 

Time complexity: O(N²)
Auxiliary Space: O(1)

Applications of Matrix Rotation:

They are useful in a variety of applications, broadly speaking, we have:-

• Image processing: In image processing, matrix rotation is often useful for rotating images to a different orientation. For example, for making an image from landscape to portrait or vice-versa.
• Data analysis: Rotating a 2D array can be useful in data analysis when visualizing or analyzing data. For example, rotating a heatmap or scatterplot can help to better understand patterns or relationships in the data.
• Computer graphics: Rotating a 2D array is useful in computer graphics for transforming objects and scenes in a 2D space. For example, rotating a 2D array of vertices can help to create animations or 2D games.

Some FAQs regarding Matrix Rotation:

1. What is the time complexity of rotating a matrix?

The time complexity of rotating a 2D matrix by 90 degrees is O(N^2), N being the number of elements in the array.

2. Can you rotate a matrix in-place?

To rotate a matrix in-place, you can perform a series of swaps on the elements in the matrix. For example, to rotate a matrix by 90 degrees, you can swap the elements in the first row with the elements in the last column, the elements in the second row with the elements in the second-to-last column, and so on. Then, you can reverse the rows of the matrix to get the final rotated matrix.

3. Can I rotate the matrix by some arbitrary angle other than multiples of 90?

No, you cannot rotate a matrix by an arbitrary angle using simple operations like swapping or reversing. To rotate a matrix by an arbitrary angle, you would need to use some more complex mathematical operations like rotation matrices or complex numbers.

4. What is the difference between clockwise and counterclockwise rotation?

Clockwise rotation means rotating the matrix to the right, while counterclockwise rotation means rotating the matrix to the left. Because of this, they are called as right and left rotation respectively.

Advantages of Rotations in a Matrix:

Rotating an array can have several advantages depending on the context in which it is used. Here are some potential benefits of rotating an array:

• Optimization: Rotating a 2D matrix can help to optimize certain algorithms or operations. For example, rotating a matrix can help to improve the performance of image processing algorithms, as it allows for better utilization of the cache memory.
• Better visualization: Rotating a 2D array can make it easier to visualize or analyze the data it contains. This can be particularly useful in data analysis, image processing, or other applications where visualization is important.
• Data encryption: Rotating a matrix can be used as a method of encryption in cryptography, where the elements of the array are replaced with their rotated positions based on a certain algorithm or key. This can help to make the data more secure.

Some Popular Interview Problems on Matrix Rotation:

• Rotate a matrix by 90 degree in clockwise direction without using any extra space
• Rotate a Matrix by 180 degree
• Rotate Matrix Elements
• Inplace rotate square matrix by 90 degrees

Some Important Problems/Article links of the same:

• Rotate a matrix by 90 degree without using any extra space | Set 2
• Rotate by 90 degree
• Rotate the matrix right by K times

Conclusion:

In conclusion, 2D array rotation is a commonly used technique in various applications, including data visualization, data analysis, image processing, and computer graphics. When we rotate a 2D array, we can gain new insights into the data or optimizing certain algorithms or operations. However, the time complexity of rotating a 2D array can be O(N^2), where N is the number of elements in the array, which can be computationally expensive for large arrays. Overall, understanding 2D array rotation and its applications can be beneficial for various data-related tasks and problem-solving.