Determinant

The Determinant

The determinant of a square matrix is a single number that encodes how the matrix scales area (2D) or volume (3D).

For a $2 imes 2$ matrix:

detegin{pmatrix} a & b \ c & d end{pmatrix} = ad - bc

def det(A):
    if len(A) == 1:
        return A[0][0]
    if len(A) == 2:
        return A[0][0]*A[1][1] - A[0][1]*A[1][0]
    # Cofactor expansion along first row
    return sum(
        ((-1)**j) * A[0][j] * det([[A[i][k] for k in range(len(A)) if k != j]
                                    for i in range(1, len(A))])
        for j in range(len(A))
    )

A = [[3, 0], [0, 2]]
print(int(round(det(A))))  # 6  — scales area by 6

What the Determinant Tells You

Singular Matrices

If rows are linearly dependent (one row is a multiple of another), the determinant is 0:

S = [[1, 2], [2, 4]]  # row 2 = 2 × row 1
print(int(round(det(S))))  # 0

Your Task

Implement det(A) that returns the determinant as an integer (rounded).