## The identity matrix

Identity matrices hold the property that, for a square matrix named A,

$IA = AI = A$

The characteristics of these matrices are that they are square and have ones along the leading diagonal, and zeros elsewhere.

## The inverse matrix A^-1

The properties of identity matrices are particularly useful in determining the inverse matrices. These matrices have the property that

$AA^{-1} = A^{-1}A = I$

In Further Mathematics, it is recommended that technology is used to find inverse matrices. However, it is also possible to do manually. Sometimes it is also discovered that not all square matrices have inverses. In order to manually calculate the inverse matrix as well as determine whether or not a matrix has an inverse, the determinant of a matrix must first be identified.

### The determinant of a matrix

The determinant of A is given by:

$det(A) = a * d - b * c$

### Calculating the inverse matrix

Given A is as above, in order to calculate the inverse $A^{-1}$,

It must be noted that in order for this to be true, det(A) cannot equal to 0. A matrix with its determinant equal to zero does not have an inverse.

### Application

Given that:

Find the inverse of B.

First of all, find the determinant a * d – b * c.
det(B) = 4*5 – 6*2 = 8

Apply the formula for the inverse. This gives: