inverse

$A^{-1}$ — the matrix with $AA^{-1}=A^{-1}A=I$ .

The inverse of a square $A$ , written $A^{-1}$ , is a matrix satisfying both $AA^{-1}=I_n$ and $A^{-1}A=I_n$ — both directions are required, since for matrices $AB=I$ does not automatically force $BA=I$ . When such a matrix exists, $A$ is invertible (or nonsingular); when it doesn't, $A$ is singular. Only square matrices can be invertible, and the inverse, when it exists, is unique.