A Python program to find the determinant of a square matrix of any size.

 

The determinant is a scalar value associated with a square matrix. It is a measure of the matrix's size, orientation, and invertibility. The determinant of a matrix is denoted by |A| or det(A), where A is the matrix.

The determinant of a 2x2 matrix A is calculated as:

Python
| A | = a11 * a22 - a12 * a21

For example, consider the following 2x2 matrix A:

Python
[ 1  2 ]
[ 3  4 ]

The determinant of A, denoted by |A| or det(A), is:

Python
| A | = 1 * 4 - 2 * 3 = -2

The determinant of a 3x3 matrix A is calculated using the following formula:

Python
| A | = a11 * (a22 * a33 - a23 * a32) - a12 * (a21 * a33 - a23 * a31) + a13 * (a21 * a32 - a22 * a31)

For example, consider the following 3x3 matrix A:

Python
[ 1  2  3 ]
[ 4  5  6 ]
[ 7  8  9 ]

The determinant of A, denoted by |A| or det(A), is:

Python
| A | = 1 * (5 * 9 - 6 * 8) - 2 * (4 * 9 - 6 * 7) + 3 * (4 * 8 - 5 * 7) = 0

A matrix is invertible (i.e., has an inverse) if and only if its determinant is non-zero. The determinant also has several other important properties, such as being distributive over matrix multiplication, and being related to the eigenvalues of the matrix, among others.

Here is a Python program to find the determinant of a square matrix of any size:

Python
def determinant(matrix):
    """
    Calculates the determinant of a square matrix using recursion.
    """
    n = len(matrix)
    if n == 1:
        return matrix[0][0]
    elif n == 2:
        return matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0]
    else:
        det = 0
        for i in range(n):
            sign = (-1) ** i
            minor = [[matrix[j][k] for k in range(n) if k != i] for j in range(1, n)]
            det += sign * matrix[0][i] * determinant(minor)
        return det

# Example usage
matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
det = determinant(matrix)
print(det) # Output: 0

The determinant function takes a square matrix as input and calculates its determinant using recursion. If the matrix is 1x1 or 2x2, the determinant is calculated directly using the formula. For larger matrices, the determinant is calculated by expanding along the first row and computing the determinants of the submatrices obtained by deleting that row and the column of each element in that row. The sign of each term in the expansion alternates according to the position of the element in the first row. The function returns the final sum of these terms, which gives the determinant of the matrix.

In the example usage, I create a 3x3 matrix and calculate its determinant using the determinant function. The output is 0, since the determinant of this particular matrix happens to be zero.

Here are some sample outputs for the given Python program to find the determinant of a square matrix:

Example 1:

python
# Matrix with size 2x2
matrix = [[1, 2], [3, 4]]
det = determinant(matrix)
print(det) # Output: -2

In this example, I calculate the determinant of a 2x2 matrix matrix using the determinant function. The output is -2, which is the correct determinant of the matrix.

Example 2:

python
# Matrix with size 3x3
matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
det = determinant(matrix)
print(det) # Output: 0

In this example, I calculate the determinant of a 3x3 matrix matrix using the determinant function. The output is 0, which is the correct determinant of the matrix.

Example 3:

python
# Matrix with size 4x4
matrix = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]
det = determinant(matrix)
print(det) # Output: 0

In this example, I calculate the determinant of a 4x4 matrix matrix using the determinant function. The output is 0, which is the correct determinant of the matrix.

Note that the determinant of a matrix is not always zero, but in Examples 2 and 3, the matrices were chosen such that their determinants happen to be zero.