Time for action – inverting matrices

The inverse of a matrix A in linear algebra is the matrix A-1, which, when multiplied with the original matrix, is equal to the identity matrix I. This can be written as follows:

A A-1 = I

The inv() function in the numpy.linalg package can invert an example matrix with the following steps:

  1. Create the example matrix with the mat() function we used in the previous chapters:
    A = np.mat("0 1 2;1 0 3;4 -3 8")
print("A
", A)

    The A matrix appears as follows:

    A
[[ 0  1  2]
 [ 1  0  3]
 [ 4 -3  8]]
  2. Invert the matrix with the inv() function:
    inverse = np.linalg.inv(A)
print("inverse of A
", inverse)

    The inverse matrix appears as follows:

    inverse of A
[[-4.5  7.  -1.5]
 [-2.   4.  -1. ]
 [ 1.5 -2.   0.5]]

    Tip

    If the matrix is singular, or not square, a LinAlgError is raised. If you want, you can check the result manually with a pen and paper. This is left as an exercise for the reader.

  3. Check the result by multiplying the original matrix with the result of the inv() function:
    print("Check
", A * inverse)

    The result is the identity matrix, as expected:

    Check
[[ 1.  0.  0.]
 [ 0.  1.  0.]
 [ 0.  0.  1.]]

What just happened?

We calculated the inverse of a matrix with the inv() function of the numpy.linalg package. We checked, with matrix multiplication, whether this is indeed the inverse matrix (see inversion.py):

from __future__ import print_function
import numpy as np

A = np.mat("0 1 2;1 0 3;4 -3 8")
print("A
", A)

inverse = np.linalg.inv(A)
print("inverse of A
", inverse)

print("Check
", A * inverse)

Pop quiz – creating a matrix

Q1. Which function can create matrices?

  1. array
  2. create_matrix
  3. mat
  4. vector

Have a go hero – inverting your own matrix

Create your own matrix and invert it. The inverse is only defined for square matrices. The matrix must be square and invertible; otherwise, a LinAlgError exception is raised.

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