Rotate a Matrix by 180 degree

Last Updated : 19 Aug, 2022

Given a square matrix, the task is that we turn it by 180 degrees in an anti-clockwise direction without using any extra space.

Examples :

Input :  1  2  3
         4  5  6
         7  8  9
Output : 9 8 7 
         6 5 4 
         3 2 1

Input :  1 2 3 4 
         5 6 7 8 
         9 0 1 2 
         3 4 5 6 
Output : 6 5 4 3 
         2 1 0 9 
         8 7 6 5 
         4 3 2 1

Method: 1 (Only prints rotated matrix)
The solution of this problem is that to rotate a matrix by 180 degrees we can easily follow that step

Matrix =  a00 a01 a02
          a10 a11 a12
          a20 a21 a22

when we rotate it by 90 degree
then matrix is
Matrix = a02 a12 a22
         a01 a11 a21
         a00 a10 a20
  
when we rotate it by again 90 
degree then matrix is 
Matrix = a22 a21 a20
         a12 a11 a10
         a02 a01 a00

From the above illustration, we get that simply to rotate the matrix by 180 degrees then we will have to print the given matrix in a reverse manner.

C++

// C++ program to rotate a matrix by 180 degrees
#include <bits/stdc++.h>
#define N 3
using namespace std;
 
// Function to Rotate the matrix by 180 degree
void rotateMatrix(int mat[][N])
{
    // Simply print from last cell to first cell.
    for (int i = N - 1; i >= 0; i--) {
        for (int j = N - 1; j >= 0; j--)
            printf("%d ", mat[i][j]);
 
        printf("\n");
    }
}
 
// Driven code
int main()
{
    int mat[N][N] = {
        { 1, 2, 3 },
        { 4, 5, 6 },
        { 7, 8, 9 }
    };
 
    rotateMatrix(mat);
    return 0;
}

Java

// Java program to rotate a
// matrix by 180 degrees
import java.util.*;
 
class GFG {
    static int N = 3;
 
    // Function to Rotate the
    // matrix by 180 degree
    static void rotateMatrix(int mat[][])
    {
 
        // Simply print from last
        // cell to first cell.
        for (int i = N - 1; i >= 0; i--) {
            for (int j = N - 1; j >= 0; j--)
                System.out.print(mat[i][j] + " ");
 
            System.out.println();
        }
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int[][] mat = { { 1, 2, 3 },
                        { 4, 5, 6 },
                        { 7, 8, 9 } };
 
        rotateMatrix(mat);
    }
}
 
// This code is contributed by ChitraNayal

Python3

# Python3 program to 
# rotate a matrix by 
# 180 degrees
N = 3;
 
# Function to Rotate 
# the matrix by 180 degree
def rotateMatrix(mat):
     
    # Simply print from
    # last cell to first cell.
    i = N - 1; 
    while(i >= 0):
        j = N - 1;
        while(j >= 0):
            print(mat[i][j], end = " ");
            j = j - 1;
        print();
        i = i - 1;
 
# Driven code
mat = [[1, 2, 3],
       [ 4, 5, 6 ],
       [ 7, 8, 9 ]];
rotateMatrix(mat);
 
# This code is contributed 
# by mits

C#

// C# program to rotate a
// matrix by 180 degrees
using System;
 
class GFG {
    static int N = 3;
 
    // Function to Rotate the
    // matrix by 180 degree
    static void rotateMatrix(int[, ] mat)
    {
        // Simply print from last
        // cell to first cell.
        for (int i = N - 1; i >= 0; i--) {
            for (int j = N - 1; j >= 0; j--)
                Console.Write(mat[i, j] + " ");
 
            Console.WriteLine();
        }
    }
 
    // Driver Code
    static public void Main()
    {
        int[, ] mat = { { 1, 2, 3 },
                        { 4, 5, 6 },
                        { 7, 8, 9 } };
 
        rotateMatrix(mat);
    }
}
 
// This code is contributed by aj_36

PHP

<?php
// PHP program to rotate 
// a matrix by 180 degree
$N = 3;
 
// Function to Rotate the
// matrix by 180 degree
function rotateMatrix($mat)
{
    global $N;
     
    // Simply print from 
    // last cell to first cell.
    for ($i = $N - 1; $i >= 0; $i--) 
    {
        for ($j = $N - 1; $j >= 0; $j--) 
            echo $mat[$i][$j], " ";
         
    echo "\n";
    }
}
 
// Driver Code
$mat = array(array(1, 2, 3),
             array(4, 5, 6),
             array(7, 8, 9));
 
rotateMatrix($mat);
 
// This code is contributed by ajit
?>

Javascript

<script>
 
// Javascript program to rotate a
// matrix by 180 degrees
N = 3;
 
// Function to Rotate the
// matrix by 180 degree
function rotateMatrix(mat)
{
     
    // Simply print from last
    // cell to first cell.
    for(var i = N - 1; i >= 0; i--) 
    {
        for(var j = N - 1; j >= 0; j--)
            document.write(mat[i][j] + " ");
 
        document.write("<br>");
    }
}
 
// Driver Code
var mat = [ [ 1, 2, 3 ],
            [ 4, 5, 6 ],
            [ 7, 8, 9 ] ];
 
rotateMatrix(mat);
 
// This code is contributed by kirti
 
</script>

Output :

 9 8 7 
 6 5 4 
 3 2 1

Time complexity: O(N*N)
Auxiliary Space: O(1)

Method : 2(In-place rotation)
There are four steps :
1- Find transpose of a matrix.
2- Reverse columns of the transpose.
3- Find transpose of a matrix.
4- Reverse columns of the transpose

Let the given matrix be
1  2  3  4
5  6  7  8
9  10 11 12
13 14 15 16

First we find transpose.
1 5 9 13
2 6 10 14
3 7 11 15
4 8 12 16

Then we reverse elements of every column.
4 8 12 16
3 7 11 15
2 6 10 14
1 5  9 13

then transpose again 
4 3 2 1 
8 7 6 5 
12 11 10 9
16 15 14 13 

Then we reverse elements of every column again
16 15 14 13 
12 11 10 9 
8 7 6 5 
4 3 2 1

C++

// C++ program for left rotation of matrix by 180
#include <bits/stdc++.h>
using namespace std;
 
#define R 4
#define C 4
 
// Function to rotate the matrix by 180 degree
void reverseColumns(int arr[R][C])
{
    for (int i = 0; i < C; i++)
        for (int j = 0, k = C - 1; j < k; j++, k--)
            swap(arr[j][i], arr[k][i]);
}
 
// Function for transpose of matrix
void transpose(int arr[R][C])
{
    for (int i = 0; i < R; i++)
        for (int j = i; j < C; j++)
            swap(arr[i][j], arr[j][i]);
}
 
// Function for display the matrix
void printMatrix(int arr[R][C])
{
    for (int i = 0; i < R; i++) {
        for (int j = 0; j < C; j++)
            cout << arr[i][j] << " ";
        cout << '\n';
    }
}
 
// Function to anticlockwise rotate matrix
// by 180 degree
void rotate180(int arr[R][C])
{
    transpose(arr);
    reverseColumns(arr);
    transpose(arr);
    reverseColumns(arr);
}
 
// Driven code
int main()
{
    int arr[R][C] = { { 1, 2, 3, 4 },
                      { 5, 6, 7, 8 },
                      { 9, 10, 11, 12 },
                      { 13, 14, 15, 16 } };
    rotate180(arr);
    printMatrix(arr);
    return 0;
}

Java

// Java program for left
// rotation of matrix by 180
import java.util.*;
 
class GFG {
    static int R = 4, C = 4, t = 0;
 
    // Function to rotate the
    // matrix by 180 degree
    static void reverseColumns(int arr[][])
    {
        for (int i = 0; i < C; i++) {
            for (int j = 0, k = C - 1; j < k; j++, k--) {
                t = arr[j][i];
                arr[j][i] = arr[k][i];
                arr[k][i] = t;
            }
        }
    }
 
    // Function for transpose of matrix
    static void transpose(int arr[][])
    {
        for (int i = 0; i < R; i++) {
            for (int j = i; j < C; j++) {
                t = arr[i][j];
                arr[i][j] = arr[j][i];
                arr[j][i] = t;
            }
        }
    }
 
    // Function for display the matrix
    static void printMatrix(int arr[][])
    {
        for (int i = 0; i < R; i++) {
            for (int j = 0; j < C; j++)
                System.out.print(arr[i][j] + " ");
            System.out.println();
        }
    }
 
    // Function to anticlockwise
    // rotate matrix by 180 degree
    static void rotate180(int arr[][])
    {
        transpose(arr);
        reverseColumns(arr);
        transpose(arr);
        reverseColumns(arr);
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int[][] arr = { { 1, 2, 3, 4 },
                        { 5, 6, 7, 8 },
                        { 9, 10, 11, 12 },
                        { 13, 14, 15, 16 } };
 
        rotate180(arr);
        printMatrix(arr);
    }
}
 
// This code is contributed by ChitraNayal

C#

// C# program for left
// rotation of matrix by 180
using System;
 
class GFG {
    static int R = 4, C = 4, t = 0;
 
    // Function to rotate the
    // matrix by 180 degree
    static void reverseColumns(int[, ] arr)
    {
        for (int i = 0; i < C; i++) {
            for (int j = 0, k = C - 1;
                 j < k; j++, k--) {
                t = arr[j, i];
                arr[j, i] = arr[k, i];
                arr[k, i] = t;
            }
        }
    }
 
    // Function for transpose of matrix
    static void transpose(int[, ] arr)
    {
        for (int i = 0; i < R; i++) {
            for (int j = i; j < C; j++) {
                t = arr[i, j];
                arr[i, j] = arr[j, i];
                arr[j, i] = t;
            }
        }
    }
 
    // Function for display the matrix
    static void printMatrix(int[, ] arr)
    {
        for (int i = 0; i < R; i++) {
            for (int j = 0; j < C; j++)
                Console.Write(arr[i, j] + " ");
            Console.WriteLine();
        }
    }
 
    // Function to anticlockwise
    // rotate matrix by 180 degree
    static void rotate180(int[, ] arr)
    {
        transpose(arr);
        reverseColumns(arr);
        transpose(arr);
        reverseColumns(arr);
    }
 
    // Driver Code
    static public void Main()
    {
        int[, ] arr = { { 1, 2, 3, 4 },
                        { 5, 6, 7, 8 },
                        { 9, 10, 11, 12 },
                        { 13, 14, 15, 16 } };
 
        rotate180(arr);
        printMatrix(arr);
    }
}
 
// This code is contributed by ajit

Python3

# Python3 program for left rotation of matrix by 180
  
R = 4
C = 4
  
# Function to rotate the matrix by 180 degree
def reverseColumns(arr):
    for i in range(C):
        j = 0
        k = C-1
        while j < k:
            t = arr[j][i]
            arr[j][i] = arr[k][i]
            arr[k][i] = t
            j += 1
            k -= 1
             
# Function for transpose of matrix
def transpose(arr):
    for i in range(R):
        for j in range(i, C):
            t = arr[i][j]
            arr[i][j] = arr[j][i]
            arr[j][i] = t
  
# Function for display the matrix
def printMatrix(arr):
    for i in range(R):
        for j in range(C):
            print(arr[i][j], end = " ");
        print();
  
# Function to anticlockwise rotate matrix
# by 180 degree
def rotate180(arr):
    transpose(arr);
    reverseColumns(arr);
    transpose(arr);
    reverseColumns(arr);
  
# Driven code
arr = [ [ 1, 2, 3, 4 ],
        [ 5, 6, 7, 8 ],
        [9, 10, 11, 12 ],
        [13, 14, 15, 16 ] ];
rotate180(arr);
printMatrix(arr);

PHP

<?php
// PHP program for left rotation of matrix by 180
 
$R = 4;
$C = 4;
  
// Function to rotate the matrix by 180 degree
function reverseColumns(&$arr)
{
    global $C;
    for ($i = 0; $i < $C; $i++)
    {
        for ($j = 0, $k = $C - 1; $j < $k; $j++, $k--)
        {
              $t = $arr[$j][$i];
              $arr[$j][$i] = $arr[$k][$i];
              $arr[$k][$i] = $t;
        }
    }    
}
  
// Function for transpose of matrix
function transpose(&$arr)
{
    global $R, $C;
    for ($i = 0; $i < $R; $i++)
    {
        for ($j = $i; $j < $C; $j++)
        {
            $t = $arr[$i][$j];
            $arr[$i][$j] = $arr[$j][$i];
            $arr[$j][$i] = $t;
        }
    }
}
  
// Function for display the matrix
function printMatrix(&$arr)
{
    global $R, $C;
    for ($i = 0; $i < $R; $i++) {
        for ($j = 0; $j < $C; $j++)
        {
            echo $arr[$i][$j]." ";
        }
        echo "\n";
    }
}
  
// Function to anticlockwise rotate matrix
// by 180 degree
function rotate180(&$arr)
{
     
    transpose($arr);
    reverseColumns($arr);
    transpose($arr);
    reverseColumns($arr);
}
  
// Driven code
 
$arr = array( array( 1, 2, 3, 4 ),
                 array( 5, 6, 7, 8 ),
                 array( 9, 10, 11, 12 ),
                 array( 13, 14, 15, 16 ) );
rotate180($arr);
printMatrix($arr);
return 0;
?>

Javascript

<script>
// Javascript program for left
// rotation of matrix by 180
 
    let  R = 4, C = 4, t = 0;
     
    // Function to rotate the
    // matrix by 180 degree
    function reverseColumns(arr)
    {
        for (let i = 0; i < C; i++) {
            for (let j = 0, k = C - 1; j < k; j++, k--) {
                t = arr[j][i];
                arr[j][i] = arr[k][i];
                arr[k][i] = t;
            }
        }
    }
     
    // Function for transpose of matrix
    function transpose(arr)
    {
        for (let i = 0; i < R; i++) {
            for (let j = i; j < C; j++) {
                t = arr[i][j];
                arr[i][j] = arr[j][i];
                arr[j][i] = t;
            }
        }
    }
     
    // Function for display the matrix
    function printMatrix(arr)
    {
        for (let i = 0; i < R; i++) {
            for (let j = 0; j < C; j++)
                document.write(arr[i][j] + " ");
            document.write("<br>");
        }
    }
     
    // Function to anticlockwise
    // rotate matrix by 180 degree
    function rotate180(arr)
    {
        transpose(arr);
        reverseColumns(arr);
        transpose(arr);
        reverseColumns(arr);
    }
     
    // Driver Code
    let arr = [ [ 1, 2, 3, 4 ],
        [ 5, 6, 7, 8 ],
        [9, 10, 11, 12 ],
        [13, 14, 15, 16 ] ];
    rotate180(arr);
    printMatrix(arr);
     
    //This code is contributed by avanitrachhadiya2155
     
</script>

Output :

 16 15 14 13 
 12 11 10 9 
 8 7 6 5 
 4 3 2 1

Time complexity : O(R*C)
Auxiliary Space : O(1)
In the code above, the transpose of the matrix has to be found twice, and also, columns have to be reversed twice.
So, we can have a better solution.

Method : 3 (Position swapping)
Here, we swap the values in the respective positions.

C++

#include <bits/stdc++.h>
using namespace std;
 
/**
 * Reverse Row at specified index in the matrix
 * @param data matrix
 * @param index row index
 */
void reverseRow(vector<vector<int>>& data,
                int index) 
{
    int cols = data[index].size();
    for(int i = 0; i < cols / 2; i++) 
    {
        int temp = data[index][i];
        data[index][i] = data[index][cols - i - 1];
        data[index][cols - i - 1] = temp;
    }
}
 
/**
 * Print Matrix data
 * @param data matrix
 */
void printMatrix(vector<vector<int>>& data)
{
    for(int i = 0; i < data.size(); i++) 
    {
        for(int j = 0; j < data[i].size(); j++) 
        {
            cout << data[i][j] << " ";
        }
        cout << endl;
    }
}
 
/**
 * Rotate Matrix by 180 degrees
 * @param data matrix
 */
void rotateMatrix180(vector<vector<int>>& data) 
{
    int rows = data.size();
    int cols = data[0].size();
 
    if (rows % 2 != 0) 
    {
         
        // If N is odd reverse the middle 
        // row in the matrix
        reverseRow(data, data.size() / 2);
    }
     
    // Swap the value of matrix [i][j] with
    // [rows - i - 1][cols - j - 1] for half     
    // the rows size. 
    for(int i = 0; i <= (rows/2) - 1; i++)
    {
        for(int j = 0; j < cols; j++) 
        {
            int temp = data[i][j];
            data[i][j] = data[rows - i - 1][cols - j - 1];
            data[rows - i - 1][cols - j - 1] = temp;
        }
    }
}
 
// Driver code    
int main()
{
    vector<vector<int>> data{ { 1, 2, 3, 4, 5 },
                              { 6, 7, 8, 9, 10 },
                              { 11, 12, 13, 14, 15 },
                              { 16, 17, 18, 19, 20 },
                              { 21, 22, 23, 24, 25 } };
 
    // Rotate Matrix
    rotateMatrix180(data);
 
    // Print Matrix
    printMatrix(data);
 
    return 0;
}
 
// This code is contributed by divyeshrabadiya07

Java

public class GFG {
       /**
     * Reverse Row at specified index in the matrix
     * @param data matrix
     * @param index row index
     */
    private static void reverseRow(int[][] data, int index) {
        int cols = data[index].length;
        for (int i = 0; i < cols / 2; i++) {
            int temp = data[index][i];
            data[index][i] = data[index][cols - i - 1];
            data[index][cols - i - 1] = temp;
        }
    }
 
    /**
     * Print Matrix data
     * @param data matrix
     */
    private static void printMatrix(int[][] data) {
        for (int i = 0; i < data.length; i++) {
            for (int j = 0; j < data[i].length; j++) {
                System.out.print(data[i][j] + " ");
            }
            System.out.println("");
        }
    }
 
    /**
     * Rotate Matrix by 180 degrees
     * @param data matrix
     */
    private static void rotateMatrix180(int[][] data) {
        int rows = data.length;
        int cols = data[0].length;
 
        if (rows % 2 != 0) {
            //If N is odd reverse the middle row in the matrix
            reverseRow(data, data.length / 2);
        }
 
        //Swap the value of matrix [i][j] with [rows - i - 1][cols - j - 1] for half the rows size. 
        for (int i = 0; i <= (rows/2) - 1; i++) {
            for (int j = 0; j < cols; j++) {
                int temp = data[i][j];
                data[i][j] = data[rows - i - 1][cols - j - 1];
                data[rows - i - 1][cols - j - 1] = temp;
            }
        }
    }
 
    public static void main(String[] args) {
        int[][] data = {
                {1, 2, 3, 4, 5},
                {6, 7, 8, 9, 10},
                {11, 12, 13, 14, 15},
                {16, 17, 18, 19, 20},
                {21, 22, 23, 24, 25}
        };
 
        //Rotate Matrix
        rotateMatrix180(data);
 
        //Print Matrix
        printMatrix(data);
    }
}

Python3

# Reverse Row at specified index in the matrix
def reverseRow(data, index):
     
    cols = len(data[index])
    for i in range(cols // 2):
        temp = data[index][i]
        data[index][i] = data[index][cols - i - 1]
        data[index][cols - i - 1] = temp
     
    return data
 
# Print Matrix data
def printMatrix(data):
     
    for i in range(len(data)):
        for j in range(len(data[0])):
            print(data[i][j], end = ' ')
         
        print()
 
# Rotate Matrix by 180 degrees
def rotateMatrix(data):
     
    rows = len(data)
    cols = len(data[0])
     
    if (rows % 2):
         
        # If N is odd reverse the middle
        # row in the matrix
        data = reverseRow(data, len(data) // 2)
         
        # Swap the value of matrix [i][j] with
        # [rows - i - 1][cols - j - 1] for half 
        # the rows size. 
        for i in range(rows // 2):
            for j in range(cols):
                temp = data[i][j]
                data[i][j] = data[rows - i - 1][cols - j - 1]
                data[rows - i - 1][cols - j - 1] = temp
         
        return data
         
# Driver Code
data = [ [ 1, 2, 3, 4, 5 ], 
         [ 6, 7, 8, 9, 10 ],
         [ 11, 12, 13, 14, 15 ],
         [ 16, 17, 18, 19, 20 ],
         [ 21, 22, 23, 24, 25 ] ]
          
# Rotate Matrix
data = rotateMatrix(data)
 
# Print Matrix
printMatrix(data)
 
# This code is contributed by rohitsingh07052

C#

using System;
 
class GFG 
{
  /**
     * Reverse Row at specified index in the matrix
     * @param data matrix
     * @param index row index
     */
  private static void reverseRow(int[, ] data, int index)
  {
    int cols = data.GetLength(1);
    for (int i = 0; i < cols / 2; i++) 
    {
      int temp = data[index, i];
      data[index, i] = data[index, cols - i - 1];
      data[index, cols - i - 1] = temp;
    }
  }
 
  /**
     * Print Matrix data
     * @param data matrix
     */
  private static void printMatrix(int[, ] data)
  {
    for (int i = 0; i < data.GetLength(0); i++) 
    {
      for (int j = 0; j < data.GetLength(1); j++) 
      {
        Console.Write(data[i, j] + " ");
      }
      Console.WriteLine();
    }
  }
 
  /**
     * Rotate Matrix by 180 degrees
     * @param data matrix
     */
  private static void rotateMatrix180(int[, ] data)
  {
    int rows = data.GetLength(0);
    int cols = data.GetLength(1);
 
    if (rows % 2 != 0) 
    {
 
      // If N is odd reverse the middle row in the
      // matrix
      reverseRow(data, data.GetLength(0) / 2);
    }
 
    // Swap the value of matrix [i][j] with [rows - i -
    // 1][cols - j - 1] for half the rows size.
    for (int i = 0; i <= (rows / 2) - 1; i++)
    {
      for (int j = 0; j < cols; j++) 
      {
        int temp = data[i, j];
        data[i, j] = data[rows - i - 1, cols - j - 1];
        data[rows - i - 1, cols - j - 1] = temp;
      }
    }
  }
 
  // Driver code
  public static void Main()
  {
    int[, ] data = { { 1, 2, 3, 4, 5 },
                    { 6, 7, 8, 9, 10 },
                    { 11, 12, 13, 14, 15 },
                    { 16, 17, 18, 19, 20 },
                    { 21, 22, 23, 24, 25 } };
 
    // Rotate Matrix
    rotateMatrix180(data);
 
    // Print Matrix
    printMatrix(data);
  }
}
 
// This code is contributed by subhammahato348

Javascript

<script>
 
// Reverse Row at specified index in the matrix
// @param data matrix
// @param index row index
function reverseRow(data,index)
{
    let cols = data[index].length;
    for(let i = 0; i < cols / 2; i++) 
    {
        let temp = data[index][i];
        data[index][i] = data[index][cols - i - 1];
        data[index][cols - i - 1] = temp;
    }
}
 
/**
 * Print Matrix data
 * @param data matrix
 */
function printMatrix(data)
{
    for(let i = 0; i < data.length; i++)
    {
        for(let j = 0; j < data[i].length; j++)
        {
            document.write(data[i][j] + " ");
        }
        document.write("<br>");
    }
}
 
/**
 * Rotate Matrix by 180 degrees
 * @param data matrix
 */
function rotateMatrix180(data)
{
    let rows = data.length;
    let cols = data[0].length;
 
    if (rows % 2 != 0) 
    {
         
        // If N is odd reverse the middle 
        // row in the matrix
        reverseRow(data, Math.floor(data.length / 2));
    }
 
    // Swap the value of matrix [i][j] 
    // with [rows - i - 1][cols - j - 1] 
    // for half the rows size.
    for(let i = 0; i <= (rows/2) - 1; i++) 
    {
        for(let j = 0; j < cols; j++) 
        {
            let temp = data[i][j];
            data[i][j] = data[rows - i - 1][cols - j - 1];
            data[rows - i - 1][cols - j - 1] = temp;
        }
    }
}
 
// Driver code
let data = [ [ 1, 2, 3, 4, 5 ],
             [ 6, 7, 8, 9, 10 ],
             [ 11, 12, 13, 14, 15 ],
             [ 16, 17, 18, 19, 20 ],
             [ 21, 22, 23, 24, 25 ] ];
// Rotate Matrix
rotateMatrix180(data);
 
// Print Matrix
printMatrix(data);
 
// This code is contributed by rag2127
    
</script>

Output :

25 24 23 22 21 
20 19 18 17 16 
15 14 13 12 11 
10 9 8 7 6 
5 4 3 2 1

Time complexity : O(R*C)
Auxiliary Space : O(1), since no extra space has been taken.

Suggest improvement

Sum of the count of number of adjacent squares in an M X N grid

Using Set() in Python Pangram Checking

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Rotate a Matrix by 180 degree

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