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Inverse Matrix

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 A

  • 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

solution:

The first step is to find the determinant of A

Next, find the inverse using the formula stated above