Computing the Determinant of a 4×4 Matrix via Cofactor Expansion

Hard

The determinant is a fundamental scalar value that can be computed from a square matrix and encapsulates essential properties of the matrix, including whether it is invertible, how it scales volumes under linear transformations, and the nature of its eigenvalues. For a 4×4 matrix, calculating the determinant involves a recursive process known as cofactor expansion (also historically referred to as Laplace expansion). This method expresses the determinant as a weighted sum of determinants of smaller submatrices called minors. The Cofactor Expansion Algorithm: For a 4×4 matrix A, the determinant can be computed by expanding along any row or column. When expanding along the first row: $$\det(A) = \sum_{j=1}^{4} (-1)^{1+j} \cdot A_{1j} \cdot \det(M_{1j})$$ Where: A₁ⱼ is the element in the first row and j-th column M₁ⱼ is the 3×3 minor matrix obtained by deleting the first row and j-th column from A (-1)^(1+j) determines the sign of each term (alternating + - + - pattern) The determinant of each 3×3 minor is then computed recursively using the same approach, expanding into 2×2 determinants. The determinant of a 2×2 matrix [[a, b], [c, d]] is simply ad - bc. Key Properties: If any row or column contains all zeros, the determinant is 0 If two rows (or columns) are identical or proportional, the determinant is 0 Swapping two rows (or columns) negates the determinant The determinant of a diagonal matrix is the product of its diagonal elements The determinant of the identity matrix is 1 Your Task: Write a Python function that computes the determinant of a 4×4 matrix using the recursive cofactor expansion method. The function should handle matrices containing integers or floating-point numbers and return the resulting determinant value.

Example 1

Input

a = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16]]

Output

Explanation

This matrix has a special structure where each row increases linearly. Because the rows are linearly dependent (each row can be expressed as a linear combination of others), the determinant equals 0. Mathematically, Row 2 - Row 1 = Row 3 - Row 2 = Row 4 - Row 3 = [4, 4, 4, 4], showing the linear dependence.

Example 2

Input

a = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]

Output

Explanation

This is the 4×4 identity matrix I₄. A fundamental property of identity matrices is that their determinant always equals 1. Using cofactor expansion: we expand along the first row where only A₁₁ = 1 is non-zero. The minor M₁₁ is the 3×3 identity matrix with determinant 1. Therefore, det(I₄) = 1 × 1 = 1.

Example 3

Input

a = [[2, 0, 0, 0], [0, 3, 0, 0], [0, 0, 4, 0], [0, 0, 0, 5]]

Output

Explanation

This is a diagonal matrix with diagonal elements 2, 3, 4, and 5. For any diagonal matrix, the determinant is simply the product of the diagonal entries: det(A) = 2 × 3 × 4 × 5 = 120 This follows from cofactor expansion since all off-diagonal elements are zero, leaving only the product of diagonal terms.

Constraints

The input matrix will always be exactly 4×4
-10⁶ ≤ a[i][j] ≤ 10⁶ for all matrix elements
Matrix elements can be integers or floating-point numbers
The result may be a very large or very small number depending on the input
Floating-point precision issues should be considered for non-integer inputs