Solution Types

Every system of linear equations falls into one of exactly three categories. Understanding which one you're dealing with before you start solving saves enormous effort — and reveals something deep about the underlying transformation.

The three cases

Unique solution — det ≠ 0

The coefficient matrix is invertible. The transformation it represents is a bijection — every output point came from exactly one input. You can always solve it: $\mathbf{x} = A^{-1}\mathbf{b}$.

No solution — det = 0, inconsistent

The matrix is singular. The transformation collapses the plane onto a line (or a point). The right-hand side $\mathbf{b}$ doesn't lie on that collapsed image, so no vector maps to it. Gaussian elimination will produce a row like $0 = k$ where $k \neq 0$.

Infinite solutions — det = 0, consistent

The matrix is singular, but $\mathbf{b}$ does lie in the image. There's an entire line of solutions — the null space of $A$shifted to one particular solution. Gaussian elimination produces a row of all zeros.

Rank: the precise measure

The rank of a matrix is the number of independent rows (or columns) it has. For a 2×2 matrix:

• Rank 2 → unique solution
• Rank 1 → either no solution or infinitely many
• Rank 0 → trivial (zero matrix)

We'll explore rank properly in the Vector Spaces module. For now: rank = how much information the matrix carries.

In code

import numpy as np

A = np.array([[1, 2], [2, 4]])   # rank 1

# Check rank
print(np.linalg.matrix_rank(A))  # 1 — singular

# np.linalg.solve will raise LinAlgError for singular A
# For least-squares (best approximate solution):
b = np.array([3, 7])
x, _, _, _ = np.linalg.lstsq(A, b, rcond=None)
print(x)  # least-squares solution