Inverse of a Non-Singular Square Matrix

Recall that a square matrix $A$ of order $n$ is non-singular if $|A|\ne0$, and singular if $|A|=0$. Matrix algebra defines scalar multiplication, addition and multiplication of matrices, but no division -- a matrix is just an arrangement of numbers, with no single numerical value to divide by.

For a non-zero real number $x$, there is always a reciprocal $y=1/x$ with $xy=yx=1$. The natural question for matrices: given a square matrix $A$, does there exist a matrix $B$ (of the same order) with

where $I$ is the identity matrix of that order? Such a $B$, when it exists, is called the inverse of $A$. …