What is the multiplicative inverse of a matrix?

2 Answers
Feb 18, 2015

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

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

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

Answer:

Just added some footnotes.

Explanation:

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

#AB = BA = I#

with #I# being the identity matrix.

This can be determined by computing the determinant of #A#.

#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.