Matrix Inverse

The Matrix Inverse

The inverse of a matrix $mathbf{A}$ is the matrix $mathbf{A}^{-1}$ such that:

$mathbf{A} cdot mathbf{A}^{-1} = mathbf{A}^{-1} cdot mathbf{A} = I$

When Does an Inverse Exist?

A matrix is invertible (non-singular) if and only if its determinant is non-zero:

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]
    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 = [[2, 1], [1, 1]]
S = [[1, 2], [2, 4]]  # det = 0 (singular)

print(abs(det(A)) > 1e-10)   # True
print(abs(det(S)) > 1e-10)   # False

Why Invertibility Matters

The equation $mathbf{A}mathbf{x} = mathbf{b}$ can be solved as $mathbf{x} = mathbf{A}^{-1}mathbf{b}$ — if $mathbf{A}$ is invertible. In practice, use Gaussian elimination instead (it's faster and more numerically stable).

Your Task

Implement is_invertible(A) that returns True if the matrix is invertible, False otherwise. Use the determinant.