Suppose $A$ and $B$ are square matrices of size $n$ such that $AB=I_n$ and $BA=I_n$. Then $A$ is invertible and $B$ is the inverse of $A$. In this situation, we write $B=\inverse{A}$.
Notice that if $B$ is the inverse of $A$, then we can just as easily say $A$ is the inverse of $B$, or $A$ and $B$ are inverses of each other.
Not every square matrix has an inverse. In Example SABMI the matrix $B$ is the inverse of the coefficient matrix of Archetype B. To see this it only remains to check that $AB=I_3$. What about Archetype A? It is an example of a square matrix without an inverse.
Let us look at one more matrix inverse before we embark on a more systematic study.
We will now concern ourselves less with whether or not an inverse of a matrix exists, but instead with how you can find one when it does exist. In Section MINM we will have some theorems that allow us to more quickly and easily determine just when a matrix is invertible.