Question

What is the multiplicative inverse of a matrix?

Answer 1

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 2

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.