How to Inverse a Matrix using NumPy

The inverse of a matrix is like the reciprocal of a number. When a matrix is multiplied by its inverse, the result is an identity matrix. It is used to solve equations and find unknown values.

The inverse of a matrix exists only if the matrix is non-singular i.e., the determinant should not be 0. Using determinant and adjoint, we can easily find the inverse of a square matrix using the below formula:

if det(A) != 0:
A_inv = adj(A) / det(A)
else:
print("Inverse doesn't exist")

Matrix Equation:

=>Ax = B\\ =>A^{-1}Ax = A^{-1}B\\ =>x = A^{-1}B

where,
A^-1: The inverse of matrix A
x: The unknown variable column
B: The solution matrix

Inverse Matrix using NumPy

numpy.linalg.inv() in the NumPy module is used to compute the inverse matrix in Python.

Syntax:

numpy.linalg.inv(a)

Parameters: a - Matrix to be inverted

Returns:  Inverse of the matrix a.

Example 1:

This example creates a 3×3 NumPy matrix and finds its inverse using np.linalg.inv()

import numpy as np
A = np.array([[6, 1, 1],
              [4, -2, 5],
              [2, 8, 7]])
print(np.linalg.inv(A))

[[ 0.17647059 -0.00326797 -0.02287582]
 [ 0.05882353 -0.13071895  0.08496732]
 [-0.11764706  0.1503268   0.05228758]]

Example 2:

This example creates a 4×4 NumPy matrix and computes its inverse using np.linalg.inv()

import numpy as np
A = np.array([[6, 1, 1, 3],
              [4, -2, 5, 1],
              [2, 8, 7, 6],
              [3, 1, 9, 7]])
print(np.linalg.inv(A))

Output:

[[ 0.13368984  0.10695187  0.02139037 -0.09090909]
 [-0.00229183  0.02673797  0.14820474 -0.12987013]
 [-0.12987013  0.18181818  0.06493506 -0.02597403]
 [ 0.11000764 -0.28342246 -0.11382735  0.23376623]]

Example 3:

This example computes the inverses of multiple NumPy matrices using np.linalg.inv()

import numpy as np
A = np.array([[[1., 2.], [3., 4.]],
              [[1, 3], [3, 5]]])
print(np.linalg.inv(A))

[[[-2.    1.  ]
  [ 1.5  -0.5 ]]

 [[-1.25  0.75]
  [ 0.75 -0.25]]]

Related Articles:

• python numpy
• matrix