Determinant of Rescaling Matrix

Theorem
Let $R$ be a commutative ring.

Let $r \in R$.

Let $r \, \mathbf I_n$ be the $n \times n$-matrix defined by:


 * $\left({r \, \mathbf I_n}\right)_{ij} = \begin{cases} r & : i = j \\ 0 & : i \ne j \end{cases}$

Then:


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

where $\det$ denotes determinant.

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


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