Notes by Lex Toumbourou

The inverse of a Matrix Transformation is a matrix that reverses the transformation.

For example, if our matrix transform did a 90° anticlockwise rotation, the inverse matrix would do a 90° clockwise rotation.

We represent the inverse of a matrix $A$ as $A^{-1}$.

When you multiply a matrix by its inverse, you get the Identity Matrix back: $A\cdot A^{-1}=I$

It's the equivalent of the reciprocal of a number in scalar math, ie $10\ast \frac{1}{10}=1$ or $10\cdot 10^{-1}=1$

For a $2\times 2$ matrix, we calculate the inverse as follows:

${\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

For example:

${\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]}^{-1}=\frac{1}{1\times 4-3\times 2}\left[\begin{array}{cc}4& -3\\ -2& 1\end{array}\right]=-\frac{1}{2}\left[\begin{array}{cc}4& -3\\ -2& 1\end{array}\right]=\left[\begin{array}{cc}-2& 1.5\\ 1& -0.5\end{array}\right]$

We can do a matrix multiplication to confirm the identity matrix is returned:

$\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]\left[\begin{array}{cc}-2& 1.5\\ 1& -0.5\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

We can also use the np.linalg.inv method in Numpy to find the inverse.

import numpy as np

m = np.array([[1, 3], [2, 4]])

m_inv = np.linalg.inv(m); m_inv

array([[-2. ,  1.5],
       [ 1. , -0.5]])

m @ m_inv

array([[1., 0.],
       [0., 1.]])

The $ad-bc$ part of the expression is the Matrix Determinate.

For a larger matrix, we can use Gaussian Elimination to invert a matrix.

Dye et al. (2018)

A matrix with a determinate of 0: $|A|=0$ is referred to as a Singular Matrix and has no inverse.

We can only calculate the inverse of a square matrix.

KhanAcademyLabs (2008)

Backlinks

• Changing Basis
• Matrix Determinate