The inverse of a matrix A is said to be the matrix which when multiplied by A results in an identity matrix. i.e.
where denotes the inverse of A
An inverse matrix has the same size as the matrix of which it is an inverse. Not all matrices have inverses. When a matrix has an inverse, it is said to be invertible. A matrix is invertible if and only if its determinant is NOT zero. The reason for this will become clear when we see how the inverse of a matrix is obtained.
Properties of Inverse Matrices
Given that matrix A is invertible, then A has the following properties:
- The determinant of A is not zero
- The determinant of the inverse of A is the inverse of the determinant of
- The inverse of an Inverse of an inverse matrix is equal to the original matrix
- The inverse of a matrix that has been multiplied by a non-zero scalar (c) is equal
to the inverse of the scalar multiplied by the inverse of the matrix
- The inverse distributes evenly across matrix multiplication
Inverse of a 2 x 2 Matrix
Given a matrix A of size 2 x 2 such that
The inverse of A can be found from the following formula:
which can also be written as
This is why a matrix with determinant zero can't have an inverse, you would end up dividing by zero!
Example: Find the inverse of A given that
The first step is to find the determinant of A
Next, find the inverse using the formula stated above