A square matrix of order is a regular (invertible) matrix if exists a matrix such that
where is an identity matrix. If a matrix is not regular, then is singular.
A matrix is unique, what we can show from the definition above.
Therefore, for a matrix we are introducing a special label: if a matrix has the inverse, that we will denote as . Now we have, by definition:
An identity matrix is the inverse of itself, that is, and zero matrix does not have an inverse matrix.
For ever two invertible matrices and of order and the identity matrix of the same order is valid:
Our mission is to explore how to determine the inverse of matrices and which matrices even have the inverse matrix.
Example 1. Find the inverse matrix of the following matrix, if exists:
We need to find a matrix such that . Let
. Therefore, we need to find numbers and in such that
Now we have
Finally, , , , , that is
We would obtain the same result in which we observed the condition .
The formula for the inverse matrix of order
For every matrix
If , then matrix do not have an inverse matrix.
An inverse matrix is a neutral element for multiplication of matrices.