inverse matrix

(Statlect) (Byjus)

Given an invertible square matrix , its inverse matrix is unique:

If , , a reciprocal of a number.

Find inverses

import numpy as np
np.linalg.inv([[1,2,3], [4,5,6], [7,8,9]])

With determinant

2x2 matrix

Given the matrix

its inverse matrix is computed using determinant

3x3 Matrix

Let

A

its inverse matrix

With elementary transformations

(Cliff Notes)

As the byproduct of Gaussian elimination

An inverse matrix is a by-product of Gaussian elimination process: it is equal to the product of all elementary matrices used to solve

There are many cases where does not exist (i.e. is invertible). But if it does, we know that

• there’s a unique solution
• the unique solution is found by

Given a linear system and is invertible, there’s a unique solution for :

So if we solve the linear system , or the augmented matrix , we can compute the inverse . See the example:

Hint

Hint

(Minerva Uni)

Special inverses

Inverse of a product

Let , its product is invertible if and only if both and are invertible

Inverse of the transpose

Let and be its transpose. If is invertible, so is and