## Determinants of Matrices over Commutative Rings

Let $$R$$ be a commutative ring. Fix an integer $$n \geq 1$$. Let $$\mathbf{A} = (a_{ij})$$ be the $$n \times n$$ square matrix

$$\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix}$$

with entry $$a_{ij}\in R$$ and $$1 \leq i,j \leq n$$ is at the intersection of the $$i^\mathrm{th}$$ row and $$j^\mathrm{th}$$ column. We will define a function $$\det(\mathbf{A}) =\det (a_{ij})$$ of this matrix $$\mathbf{A}$$, which will also be denoted by

$$\begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{vmatrix}$$

We will also state and prove the various familiar properties of the determinant in this setting.

© 2015 George S. of www.notionsandnotes.org.