Inverse of a Matrix

A square matrix $A$ of order $n$ is called invertible if there exists another square matrix $B$ of the same order such that $AB = BA = I$, where $I$ is the identity matrix of order $n$. When such a $B$ exists, it is called the inverse of $A$, written $A^{-1}$ — and it is always unique.

The inverse behaves in ways that echo reciprocals of ordinary numbers: for two invertible matrices $A$ and $B$ of the same order, $(AB)^{-1} = B^{-1}A^{-1}$ (note the order reverses); taking the inverse twice returns the original matrix, $(A^{-1})^{-1} = A$; and the inverse of a transpose equals the transpose of the inverse, $(A^{T})^{-1} = (A^{-1})^{T}$. …