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Linear Algebra for Data Science Using Python

Inverse of a Square Matrix

Explore the concept of the inverse of a square matrix, how to calculate it using Python and Gauss-Jordan elimination, and its applications in data science. Understand criteria for invertibility, singularity, and how inverses help solve linear equations and regenerate matrices.

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Definition

The inverse of a square matrix AA is another matrix, A−1A^{-1} of the same size that converts AA to identity, II, upon multiplication. Mathematically,

A−1A=AA−1=IA^{-1}A = AA^{-1}= I

The matrix, A−1A^{-1}, has an opposite effect to that of AA. Whatever AA does, A−1A^{-1} undoes.

The Python code to calculate the inverse of a square matrix is as follows:

Python 3.8
import numpy as np
mat = np.array([[1, 0, 0],
                [0, 1, 0],
                [0, 0, 2]])
print(mat)
inv = np.linalg.inv(mat)    
print(f"""The inverse of the matrix is\n{inv}""")

Example

For example, the elimination matrix EE represents two operations, r3=r3+2r2\bold{r}_3 = \bold{r}_3 + 2\bold{r}_2 and r1=3r1\bold{r}_1 = 3\bold{r}_1:

E=[300010021]E=\begin{bmatrix} 3 & 0 & 0\\ 0 & 1 & 0\\0 & 2 & 1 \end{bmatrix}

The inverse of this matrix, E−1E^{-1} , is a matrix that does the reverse operations, r3=r3−2r2\bold{r}_3 = \bold{r}_3 - 2\bold{r}_2 and r1=13r1\bold{r}_1 = \frac{1}{3}\bold{r}_1 ...