3. Inverting matrices

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

– John von Neumann (1903 - 1957)

If two square matrices $A$ and $B$ are such that $A\cdot B$ and $B\cdot A$ are both equal to the identity matrix $I_n$, then they are called inverses of each other. In this section we learn how to invert square matrices of any size by using row operations. In comparison to the formulae that you may have learnt for inverting $2\times 2$ and $3\times 3$ matrices, this method is better because it requires less memorization, and it can be used with arbitrarily large matrices.