Let $A$ and $B$ be arbitrary $n \times n$ matrices. If $B$ is invertible, prove that $AB^{-1} = B^{-1}A$ if and only if $AB=BA$.

Question

Let $A$ and $B$ be arbitrary $n \times n$ matrices. If $B$ is invertible, prove that $AB^{−1} = B^{−1}A$ if and only if $AB=BA$. Would appreciate a solution to this proof.

I'm confused whether my proof should be showing how $AB^{-1} = B^{−1}A$ or I should prove $AB = BA$ from $AB^{−1} =B^{−1}A$ or prove that $AB^{−1} =B^{−1}A$ from $AB=BA$.

And if so, should I use $\Longrightarrow$ sign for each new line or $\Longleftrightarrow$ (given that this sign means if and only if)?

Answer 1

To prove ,$AB^{-1}=B^{-1}A$ iff $AB=BA$

$AB^{-1}=B^{-1}A$

$\Leftrightarrow$ $BAB^{-1}B=BB^{-1}AB$

$\Leftrightarrow$ $BAI=IAB$

$\Leftrightarrow$ $BA=AB$

$\Leftrightarrow$ $AB=BA$