Matrix inversion
Recover an inverse when a square matrix is known to be nonsingular.
NumPy · Function
np.linalg.inv
Computes the inverse of a square matrix, returning a result that multiplies with the input to form the identity. It raises LinAlgError when the input is nonsquare or inversion fails.
How it works
The function accepts a square matrix, including batches whose final two dimensions are square, and returns an inverse with matching matrix dimensions. The inverse is the array that produces the identity when multiplied with the input in either order; failure to invert raises LinAlgError.
Syntax
numpy.linalg.inv(a)Parameters
| Name | Type | Description |
|---|---|---|
a | (..., M, M) array_like | Matrix to be inverted; expected shape (..., M, M). |
Returns
ainv (..., M, M) ndarray or matrix: inverse of the matrix a.
Key points
- For a square matrix, it produces the multiplicative inverse.
- The result is defined by identity-matrix multiplication on both sides.
- The input can be any array-like object whose final two dimensions form an M×M matrix.
- The output preserves the matrix dimensions and is returned as an ndarray or matrix.
- Singularity is explicitly checked as a condition that prevents inversion.
Common mistakes
- Singular matrices cannot be inverted; handle numpy.linalg.LinAlgError when the input has no inverse.
- The operation also raises LinAlgError for nonsquare input or another inversion failure.
Example: Invert a square matrix
import numpy as np
A = np.array([[1.0, 2.0], [3.0, 4.0]])
A_inv = np.linalg.inv(A)
print(np.allclose(A @ A_inv, np.eye(2)))
print(np.allclose(A_inv @ A, np.eye(2)))Computes the inverse and verifies both multiplication orders produce the 2×2 identity matrix.
When to use it
Batched inversion
Process several matrices together when each item occupies the final two axes.
Numerical validation
Verify an inversion numerically by checking products against an identity matrix.
Interview questions
QWhat does numpy.linalg.inv compute?
The function returns an array ainv such that A @ ainv and ainv @ A equal the identity matrix. Its input must have square final dimensions.
Follow-up: How can you validate the result? Check both multiplication orders against np.eye using np.allclose, while also handling LinAlgError.
QWhat shapes does inv accept and return?
The parameter has shape (..., M, M), so the input can represent one square matrix or a stack of square matrices. The returned value has shape (..., M, M).
Follow-up: What shape constraint applies to each item in a batch? Every matrix represented by the final two axes must be M by M.
QWhich errors can inv raise?
It raises numpy.linalg.LinAlgError when the input is not square or inversion fails. In particular, a detected singular matrix causes this exception.
Follow-up: What should code do for a singular matrix? Use a try/except around inv or choose a method suited to the problem, such as a solve operation when solving a system.
Related
numpy.linalg.solvenumpy.linalg.pinvnumpy.linalg.detnumpy.linalg.slogdetnumpy.linalg.matrix_power