How to Solve Linear Equations Using NumPy in Python

To solve linear equations in numpy, use numpy.linalg.solve(A, b) where A is the coefficient matrix and b is the constants vector. This function returns the solution vector that satisfies the equation Ax = b.

Why This Happens

When trying to solve linear equations, a common mistake is to use incorrect shapes for the coefficient matrix or the constants vector. For example, passing a 1D array instead of a 2D matrix for coefficients or mismatched dimensions causes errors.

python

import numpy as np

A = np.array([1, 2, 3, 4])  # Incorrect shape, should be 2D
b = np.array([5, 6])

solution = np.linalg.solve(A, b)

Output

ValueError: expected square matrix

The Fix

Make sure the coefficient matrix A is a square 2D array and the constants vector b has a matching number of rows. Use numpy.linalg.solve(A, b) to get the solution vector.

python

import numpy as np

A = np.array([[1, 2], [3, 4]])  # 2x2 matrix
b = np.array([5, 6])            # vector with 2 elements

solution = np.linalg.solve(A, b)
print(solution)

Output

[-4.   4.5]

Prevention

Always check that your coefficient matrix is square (same number of rows and columns) and that your constants vector matches the number of rows. Use array.shape to verify dimensions before solving. Avoid singular matrices (matrices without an inverse) as they cause errors.

Use numpy.linalg.cond(A) to check if the matrix is well-conditioned (a low value means safe to solve).

Related Errors

LinAlgError: Singular matrix: This happens when the coefficient matrix cannot be inverted. Check if your equations are dependent or if the matrix is not full rank.

ValueError: shapes not aligned: This occurs when the dimensions of A and b do not match. Always verify shapes before solving.

Key Takeaways

Use numpy.linalg.solve with a square coefficient matrix and matching constants vector.

Check array shapes with .shape before solving linear equations.

Avoid singular matrices to prevent errors during solving.

Use numpy.linalg.cond to check matrix condition before solving.

Match dimensions exactly: A must be NxN and b must be length N.