# What is the multiplicative inverse of a matrix?

Feb 18, 2015

The multiplicative inverse of a matrix $A$ is a matrix (indicated as ${A}^{-} 1$) such that:
$A \cdot {A}^{-} 1 = {A}^{-} 1 \cdot A = I$
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]

Jul 23, 2015

#### Explanation:

Firstly, the matrix described here needs to be square $\left(n \times n\right)$ and invertible, such that for a given square matrix $A$, there exists a square matrix $B$ where

$A B = B A = I$

with $I$ being the identity matrix.

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

$A = \left(\begin{matrix}a & b \\ c & d\end{matrix}\right)$

The determinant of $A$, $\det \left(A\right)$, will be

$\det \left(A\right) = a d - b c$

If $\det \left(A\right) = 0$, $A$ is singular (opposite of invertible) ${A}^{-} 1$ doesn't exist, but if

$\det \left(A\right) \ne 0$, $A$ is invertible and ${A}^{-} 1$ exists.