Rotate a given n × n matrix by 90 degrees in the clockwise direction without using an additional matrix.

Question

Accepted Answer

Approach:
The optimal in-place solution consists of two steps:

1. Transpose the matrix by swapping elements across its main diagonal. This converts rows into columns.
2. Reverse each row of the transposed matrix to achieve a 90-degree clockwise rotation.

This approach avoids using extra memory for another matrix and modifies the original matrix directly, making it suitable for large inputs and interview scenarios.

Java Solution:
```
public class RotateImage {

public static void rotate(int[][] matrix) {
        int size = matrix.length;

// Step 1: Transpose the matrix
        for (int row = 0; row < size; row++) {
            for (int column = row + 1; column < size; column++) {
                int temp = matrix[row][column];
                matrix[row][column] = matrix[column][row];
                matrix[column][row] = temp;
            }
        }

// Step 2: Reverse each row
        for (int row = 0; row < size; row++) {
            int left = 0;
            int right = size - 1;

while (left < right) {
                int temp = matrix[row][left];
                matrix[row][left] = matrix[row][right];
                matrix[row][right] = temp;

left++;
                right--;
            }
        }
    }

public static void printMatrix(int[][] matrix) {
        for (int[] row : matrix) {
            for (int value : row) {
                System.out.print(value + " ");
            }
            System.out.println();
        }
    }

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

rotate(matrix);

printMatrix(matrix);
    }
}
```
Output:
7 4 1
8 5 2
9 6 3

Time Complexity:
O(n²), where n is the dimension of the matrix. Every element is visited a constant number of times during transpose and reversal operations.

Space Complexity:
O(1), since the rotation is performed in-place without allocating another matrix.

Key Interview Points:
• The transpose-and-reverse technique is the standard optimal solution for in-place matrix rotation.
• Using an additional matrix would simplify implementation but increase space complexity to O(n²).
• A common interview follow-up is to rotate the matrix by 90 degrees anti-clockwise or by 180 degrees.