linear algebra - Inverse Matrix and matrix multiplication

If I got the invertible matrix $A$, I can calculate the inverse matrix $A^{-1}$, so that $A \cdot A^{-1} = E$, where $E$ is the identity matrix.

Wikipedia says that not only $A\cdot A^{-1} = E$ must be fulfilled, but also $A^{-1} \cdot A = E $. Can someone explain to me why this is not a contradiction to the fact that matrix multiplication is not commutative ? Is the inverse matrix really defined as a matrix which fulfills both?

Answer

The inverse of a matrix is defined as the matrix that satisfies both relationships.

For square matrices $A$ and $B$,
$$
B\mbox{ is the inverse of }A:=B\mbox{ such that } AB{}={}BA{}={}I\,.
$$

Incidentally, this also means that $A$ is the inverse of $B$.