Determinant of Rescaling Matrix

Theorem
Let $R$ be a commutative ring.

Let $r \in R$, and let $r \, \mathbf{I}_n$ be the $n \times n$-matrix defined by:


 * $\left({r \, \mathbf{I}_n}\right)_{ij} = \begin{cases}r & \text{if $i = j$}\\0 & \text{if $i \ne j$}\end{cases}$

Then:


 * $\det \left({r \, \mathbf{I}_n}\right) = r^n$

where $\det$ denotes determinant.

Proof
From Determinant of a Diagonal Matrix, it follows directly that:


 * $\det \left({r \, \mathbf{I}_n }\right) = \displaystyle \prod_{i \mathop = 1}^n r = r^n$