What is the multiplicative inverse of a matrix?

2 Answers
Feb 18, 2015

The multiplicative inverse of a matrix AA is a matrix (indicated as A^-1A1) such that:
A*A^-1=A^-1*A=IAA1=A1A=I
Where II is the identity matrix (made up of all zeros except on the main diagonal which contains all 11).
For example:
if: A=A=
[4 3]
[3 2]

A^-1=A1=
[-2 3]
[3 -4]

Try to multiply them and you'll find the identity matrix:
[1 0]
[0 1]

Just added some footnotes.

Explanation:

Firstly, the matrix described here needs to be square (n xx n)(n×n) and invertible, such that for a given square matrix AA, there exists a square matrix BB where

AB = BA = IAB=BA=I

with II being the identity matrix.

This can be determined by computing the determinant of AA.

A = ( (a,b), (c,d))

The determinant of A, det(A), will be

det(A)= ad - bc

If det(A) = 0, A is singular (opposite of invertible) A^-1 doesn't exist, but if

det(A) != 0, A is invertible and A^-1 exists.