Inverting a matrix is an advanced interview topic. Today, I am going to explain the steps to reverse a matrix or flip a matrix in C#.

Greetings, Interview Preparation Enthusiasts! Today, we're delving into the realm of matrix inversion, a fundamental concept in linear algebra with significant applications in engineering, physics, and computer science. Join us as we thoroughly examine matrix inversion, accompanied by a practical C# implementation example designed to help you tackle your upcoming coding challenges.

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public static double[,] InvertMatrix(double[,] matrix)
{
    int n = matrix.GetLength(0);
    double[,] augmented = new double[n, n * 2];

    // Initialize augmented matrix with the input matrix and the identity matrix
    for (int i = 0; i < n; i++)
    {
        for (int j = 0; j < n; j++)
        {
            augmented[i, j] = matrix[i, j];
            augmented[i, j + n] = (i == j) ? 1 : 0;
        }
    }

    // Apply Gaussian elimination
    for (int i = 0; i < n; i++)
    {
        int pivotRow = i;
        for (int j = i + 1; j < n; j++)
        {
            if (Math.Abs(augmented[j, i]) > Math.Abs(augmented[pivotRow, i]))
            {
                pivotRow = j;
            }
        }

        if (pivotRow != i)
        {
            for (int k = 0; k < 2 * n; k++)
            {
                double temp = augmented[i, k];
                augmented[i, k] = augmented[pivotRow, k];
                augmented[pivotRow, k] = temp;
            }
        }

        if (Math.Abs(augmented[i, i]) < 1e-10)
        {
            return null;
        }

        double pivot = augmented[i, i];
        for (int j = 0; j < 2 * n; j++)
        {
            augmented[i, j] /= pivot;
        }

        for (int j = 0; j < n; j++)
        {
            if (j != i)
            {
                double factor = augmented[j, i];
                for (int k = 0; k < 2 * n; k++)
                {
                    augmented[j, k] -= factor * augmented[i, k];
                }
            }
        }
    }

    double[,] result = new double[n, n];
    for (int i = 0; i < n; i++)
    {
        for (int j = 0; j < n; j++)
        {
            result[i, j] = augmented[i, j + n];
        }
    }

    return result;
}

Step 1 : Matrix Initialization

Initialize the matrix as a 2D array.
Example: A 4x4 matrix with predefined values.

Step 2 : Augmented Matrix Setup

Create an augmented matrix combining the original matrix with an identity matrix.
Example: The left half is the original matrix, and the right half is the identity matrix.

Step 3 : Gaussian Elimination Process

Transform the matrix to its reduced row echelon form by selecting the pivot element and swapping rows if necessary.
Example: Ensure the pivot element is the largest absolute value in the column.

Step 4 : Row Normalization

Normalize the pivot row so that the pivot element becomes 1.
Example: Divide each element of the pivot row by the pivot element.

Step 5 : Column Elimination

Eliminate the other entries in the current column to zero out elements above and below the pivot.
Example: Subtract the appropriate multiple of the pivot row from each other row.

Step 6 : Extracting and Returning the Inverse

Extract the inverse matrix from the augmented matrix.
Example: Copy the right half of the augmented matrix into a separate matrix for the inverse.