Inverse of Rescaling Matrix

Theorem

Let $R$ be a commutative ring with unity.

Let $r \in R$ be a unit in $R$.

Let $r \, \mathbf I_n$ be the $n \times n$ rescaling matrix of $r$.

Then:

$\paren {r \, \mathbf I_n}^{-1} = r^{-1} \, \mathbf I_n$

Proof

By definition, a rescaling matrix is also a diagonal matrix.

Hence Inverse of Diagonal Matrix applies, and since $r$ is a unit, it gives the desired result.

$\blacksquare$