numpy.linalg.LinAlgError: Singular matrix (tested causes and fixes)

numpy.linalg.LinAlgError: Singular matrix means np.linalg.inv or solve got a square matrix with linearly dependent rows (determinant 0), so there is no unique answer. In practice it is usually a duplicated or collinear column in a regression. Below: how to confirm it, why nearly singular matrices slip through, and the fixes, tested on NumPy 2.5.4.

1. The error

import numpy as np

A = np.array([[1.0, 2.0], [2.0, 4.0]])   # row 2 = 2 * row 1
print("det:", np.linalg.det(A), "rank:", np.linalg.matrix_rank(A))
try:
    np.linalg.inv(A)
except np.linalg.LinAlgError as e:
    print("LinAlgError:", e)
try:
    np.linalg.solve(A, np.array([1.0, 2.0]))
except np.linalg.LinAlgError as e:
    print("LinAlgError:", e)

Output:

det: 0.0 rank: 1
LinAlgError: Singular matrix
LinAlgError: Singular matrix

A square matrix is singular when its rows (or columns) are linearly dependent: here row 2 is twice row 1, so the determinant is 0 and the rank is 1 instead of 2. Such a matrix has no inverse, and A x = b has either no solution or infinitely many, so both inv and solve raise LinAlgError: Singular matrix. Check np.linalg.matrix_rank(A) to confirm: rank below the size means singular.

2. Nearly singular matrices do not raise

import numpy as np

B = np.array([[1.0, 2.0], [1.0, 2.0 + 1e-15]])   # nearly singular
print("det:", np.linalg.det(B), "cond:", np.linalg.cond(B))
x = np.linalg.solve(B, np.array([1.0, 2.0]))     # no error...
print("x:", x)                                   # ...but huge, meaningless values

Output:

det: 8.881784197001244e-16 cond: 1.5922629181314436e+16
x: [-2.25179981e+15  1.12589991e+15]

A matrix that is singular only up to rounding error passes without an error, but the result is garbage: the condition number is about 1.6e16 and the solution is in the 1e15 range. If np.linalg.cond(A) is near 1/np.finfo(float).eps (about 4.5e15), treat the matrix as singular. The fact that an error appears or not is therefore not a reliable test; a tiny change in the data can flip it.

3. Regression: duplicate or collinear columns

import numpy as np

# regression design matrix with a duplicated column
X = np.column_stack([np.ones(4), [1, 2, 3, 4], [2, 4, 6, 8]]).astype(float)
y = np.array([3.0, 5.0, 7.0, 9.0])
try:
    np.linalg.inv(X.T @ X) @ X.T @ y
except np.linalg.LinAlgError as e:
    print("LinAlgError:", e)

coef, res, rank, sv = np.linalg.lstsq(X, y, rcond=None)
print("lstsq:", coef.round(4), "rank:", rank)
print("pinv: ", (np.linalg.pinv(X) @ y).round(4))

Output:

LinAlgError: Singular matrix
lstsq: [1.  0.4 0.8] rank: 2
pinv:  [1.  0.4 0.8]

The most common real-world cause: a design matrix with a column that is a multiple of another (here column 3 = 2 × column 2), a constant column plus a full set of one-hot dummies, or more columns than rows. X.T @ X is then singular. Fix the data by dropping the redundant column (or one dummy per category, pd.get_dummies(..., drop_first=True)), or use np.linalg.lstsq / np.linalg.pinv, which return the minimum-norm least-squares solution and report the rank (2 here, not 3).

4. Other cases

import numpy as np

C = np.zeros((2, 2))
try:
    np.linalg.inv(C)
except np.linalg.LinAlgError as e:
    print("LinAlgError:", e)
try:
    np.linalg.inv(np.ones((2, 3)))
except np.linalg.LinAlgError as e:
    print("LinAlgError:", e)
print(np.linalg.inv(np.array([[1, 2], [3, 4]])))   # ints are fine

Output:

LinAlgError: Singular matrix
LinAlgError: Last 2 dimensions of the array must be square
[[-2.   1. ]
 [ 1.5 -0.5]]

An all-zero matrix (often a bug upstream: an empty selection or a wrong variable) is singular too. A non-square matrix gives a different LinAlgError: "Last 2 dimensions of the array must be square"; use pinv or lstsq for those. Integer matrices are converted to float automatically and are not a cause.

5. Checklist

• Print A.shape, np.linalg.matrix_rank(A) and np.linalg.cond(A).
• Look for duplicate, constant or all-zero rows/columns.
• Prefer np.linalg.solve(A, b) over inv(A) @ b, and lstsq for regression.
• Use pinv only when you accept a minimum-norm answer to an underdetermined problem.

Related: NumPy RuntimeWarning (divide by zero, overflow) · assignment destination is read-only · numpy has no attribute 'int'.

Written with AI assistance. Measured on 2026-10-23 in Docker: python:3-alpine (Python 3.14.7) with numpy 2.5.4. Each block was run verbatim and the output pasted unedited. PlainTables is our extension; Python and NumPy are free.