Let $A$ and $B$ be two square matrices with the same dimensions. If the result of multiplying $A$ by $B$ from either side is the identity matrix $I,$ then $B$ is the multiplicative inverse of $A.$

The multiplicative inverse of a matrix $A$ is usually denoted $A^{\text{-}1}.$

Example

Consider two $3\times 3$ matrices. It can be shown that $B$ is the multiplicative inverse of $A$ by calculating the products $A\cdot B$ and $B\cdot A.$ First, the product $A\cdot B$ is calculated. Next, the product $B\cdot A$ is calculated. Since $A\cdot B=I$ and $B\cdot A=I,$ $B$ is the multiplicative inverse of $A$ and $B=A^{\text{-}1}.$