How to invert a matrix or nArray in Python?

In this article, we will show you how to calculate the inverse of a matrix or ndArray using NumPy library in python.

What is inverse of a matrix?

The inverse of a matrix is such that if it is multiplied by the original matrix, it results in an identity matrix.

The inverse of a matrix is just the reciprocal of the matrix, as in regular arithmetic, for a single number that is used to solve equations to obtain the value of unknown variables. The inverse of a matrix is the matrix that, when multiplied by the original matrix, produces the identity matrix.

The inverse of a matrix exists only if the matrix is non-singular, that is, if the determinant is not 0. We can simply find the inverse of a square matrix using the determinant and adjoint using the formula below

if det(A) != 0
 A-1 = adj(A)/det(A)
else
 "Inverse does not exist"

Method 1 − Using numpy.linalg.inv() function for np.array() type

numpy.linalg.inv() function

Python has a very simple method for calculating the inverse of a matrix. To compute the inverse of a matrix, use the numpy.linalg.inv() function from the NumPy module in Python bypassing the matrix.

Syntax

numpy.linalg.inv(array)

Parameters

array − It is the matrix that must be inverted.

Return Value − numpy.linalg.inv() function returns the inverse of a matrix.

Algorithm (Steps)

Following are the Algorithm/steps to be followed to perform the desired task −

• Use the import keyword, to import the numpy module with an alias name(np).

• Use the numpy.array() function(returns a ndarray. The ndarray is an array object that satisfies the given requirements), for creating a numpy array by passing the 3-Dimensional array(3rows, 3columns) as an argument to it.

• Use the linalg.inv() function(calculates the inverse of a matrix) of the numpy module to calculate the inverse of an input 3x3 matrix by passing the input matrix as an argument to it and print the inverse matrix.

Example

The following program returns the inverse of an input 3-Dimensional(3x3) matrix using the numpy.linalg.inv() function −

# importing numpy module with an alias name
import numpy as np

# creating a 3-Dimensional(3x3) numpy matrix
inputArray_3d = np.array([[4, 5, 1],
   [3, 4, 12],
   [10, 2, 1]])

# printing the input 3D matrix
print("The input numpy 3D matrix:")
print(inputArray_3d)

# calculating the inverse of an input 3D matrix
resultInverse= np.linalg.inv(inputArray_3d)

# printing the resultant inverse of an input matrix
print("The Inverse of 3-Dimensional(3x3) numpy matrix:")
print(resultInverse)

Output

On executing, the above program will generate the following output −

The input numpy 3D matrix:
[[ 4  5  1]
 [ 3  4 12]
 [10  2  1]]
The Inverse of 3-Dimensional(3x3) numpy matrix:
[[-0.04246285 -0.00636943  0.11889597]
 [ 0.24840764 -0.01273885 -0.0955414 ]
 [-0.07218684  0.08917197  0.00212314]]

Method 2 − Using scipy.linalg.inv() function

scipy.linalg.inv()

Using the scipy module's functionalities, we can perform various scientific calculations. It also works with numpy arrays.

In Python, scipy.linalg.inv() can also return the inverse of a given square matrix. It works in the same way as the numpy.linalg.inv() function.

Algorithm (Steps)

Following are the Algorithm/steps to be followed to perform the desired task −

• Use the import keyword, to import linalg from scipy module.

• Use the numpy.matrix() function(returns a matrix from a string of data or an array-like object. The resulting matrix is a specialized 2D array), for creating a numpy matrix by passing the 2-Dimensional array(2rows, 2columns) as an argument to it.

• Use the linalg.inv() function(calculates the inverse of a matrix) of the scipy module to calculate the inverse of an input 2x2 matrix by passing the input matrix as an argument to it and print the inverse matrix.

Example

import numpy as np
# importing linalg from scipy module
from scipy import linalg

# creating a 2-Dimensional(2x2) NumPy matrix
inputMatrix = np.matrix([[5, 2],[7, 3]])

# printing the input 2D matrix
print("The input numpy 2D matrix:")
print(inputMatrix)

# calculating the inverse of an input 2D matrix
resultInverse = linalg.inv(inputMatrix)

# printing the resultant inverse of an input matrix
print("The Inverse of 2-Dimensional(2x2) numpy matrix:")
print(resultInverse)

Output

The input numpy 2D matrix:
[[5 2]
[7 3]]
The Inverse of 2-Dimensional(2x2) numpy matrix:
[[ 3. -2.]
[-7. 5.]]

Method 3 − Using numpy.linalg.inv() function for np.matrix() type

Algorithm (Steps)

Following are the Algorithm/steps to be followed to perform the desired task −

• Use the numpy.matrix() function(returns a matrix from a string of data or an array-like object. The resulting matrix is a specialized 4D array), for creating a numpy matrix by passing the 4-Dimensional array(4 rows, 4 columns) as an argument to it.

Example

import numpy as np

# creating a NumPy matrix (4x4 matrix) using matrix() method
inputMatrix = np.matrix('[11, 1, 8, 2; 11, 3, 9 ,1; 1, 2, 3, 4; 9, 8, 7, 6]')

# printing the input 4D matrix
print("The input NumPy matrix:")
print(inputMatrix)

# calculating the inverse of an input matrix
resultInverse= np.linalg.inv(inputMatrix)

# printing the resultant inverse of an input matrix
print("The Inverse of 4-Dimensional(4x4) numpy matrix:")
print(resultInverse)

Output

The input NumPy matrix:
[[11 1 8 2]
[11 3 9 1]
[ 1 2 3 4]
[ 9 8 7 6]]
The Inverse of 4-Dimensional(4x4) numpy matrix:
[[ 0.25   -0.23214286   -0.24107143   0.11607143]
[-0.25     0.16071429   -0.09464286   0.11964286]
[-0.25     0.375         0.3125      -0.1875    ]
[ 0.25    -0.30357143    0.12321429   0.05178571]]

Conclusion

In this article, we learned how to calculate the inverse of a matrix using three different examples. We learned how to take a matrix in Numpy using two different methods:numpy.array() and NumPy.matrix().

Vikram Chiluka

Updated on: 31-Oct-2022

18K+ Views

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