How to Inverse a Matrix using Numpy

A matrix inverse is a matrix that, when multiplied with the original matrix, results in an identity matrix.

In linear algebra, the inverse of a matrix is an important concept that is widely used in various applications.

The inverse of a matrix is denoted as A^-1, where A is the original matrix.

In this blog post, we will discuss how to calculate the inverse of a matrix using Numpy, a popular Python library for numerical computing.

Here are the steps to calculate the inverse of a matrix using Numpy:

Import the Numpy library:

import numpy as np

Define the matrix:

A = np.array([[a, b], [c, d]])

Use the numpy.linalg.inv() function to calculate the inverse of the matrix:

A_inv = np.linalg.inv(A)

That's it! The numpy.linalg.inv() function returns the inverse of the matrix A.

Example 1

import numpy as np

# Define the matrix
A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

# Calculate the inverse of the matrix
A_inv = np.linalg.inv(A)

# Display the inverse matrix
print("Inverse of the matrix:\n", A_inv)

The output will be:

Inverse of the matrix:
 [[-0.16666667  0.33333333 -0.16666667]
 [ 0.33333333 -0.5        0.16666667]
 [-0.16666667  0.16666667  0.16666667]]

Note that the inverse matrix is not always unique, and its values may vary depending on the implementation and the library used.

However, the property of the inverse matrix remains the same - it should satisfy the property of A * A^-1 = I, where I is the identity matrix.

Example 2

Here's another example of how to calculate the inverse of a matrix using Numpy:

import numpy as np

# Define the matrix
A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

# Calculate the inverse of the matrix
A_inv = np.linalg.inv(A)

# Verify the result by multiplying A with its inverse
result = np.dot(A, A_inv)

# Round off the result to 2 decimal places
result = np.round(result, 2)

# Display the result
print("Result: \n", result)

The output will be:

Result: 
 [[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]

In this example, we first defined the matrix A and then used the numpy.linalg.inv() function to calculate its inverse.

Next, we verified the result by multiplying A with its inverse.

The result should be an identity matrix, which means that the inverse is correct.

In this example, we rounded off the result to 2 decimal places using the numpy.round() function to make the output easier to read.

It's important to note that not all matrices have an inverse.

A matrix can only have an inverse if it is square and has a non-zero determinant.

The numpy.linalg.inv() function raises a LinAlgError exception if the matrix does not have an inverse.

Summary

In conclusion, Numpy makes it easy to calculate the inverse of a matrix.

By following these simple steps, you can quickly find the inverse of a matrix in your Python code.