Question on the definition of an Inverse matrix

Question

By definition, if $A$ is a $ n \times n $ matrix, an inverse of $A$ is an $ n \times n $ matrix $A^{-1}$ with the property that:

$$ A^{-1}A=\mathbb I_n \ \ \land \ \ AA^{-1}=\mathbb I_n \ \ \ \ (1)$$

where $ \mathbb I_n $ is the $ n \times n $ identity matrix.

Are there any cases where $ A^{-1}A=\mathbb I_n$ but $AA^{-1} \neq \mathbb I_n$ or the other way around (and thus making (1) a false statement) ?

Answer 1

Let $A$ be any invertible matrix in $\mathbb R^{n\times n}$. We define $B$ to be the left-hand inverse of $A$ satisfying $BA=I_n$ and $C$ to be the right hand inverse of $A$ satisfying $AC=I_n$.

Now, observe that $$ B = BI_n = B(AC) = (BA)C = I_nC = C. $$

Hence, the left-hand and right-hand inverses of all $n\times n$ invertible matrix are always equal.