Cofactor Expansion of the Determinant along a Row or a Column, and the Adjugate Identity

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $n\in\mathbb{N}$ with $2\le n$ for the order of $\mathbb{N}_{0}$, and let $A\in\mathcal{M}_{n}(\mathbb{R})$ be a square real matrix with entries $A_{ij}$, with the determinant $\det$, the cofactors $C_{ij}(A)$ and the adjugate $\operatorname{adj}(A)$ of $A$, the matrix product, the scalar multiple of a matrix, the identity matrix $I_{n}$, and sums as in Commutative Rings, Fields and Ordered Fields: Standard Notation §rings.

¶ For every $i\in[n]$, $\det A=\sum_{j=1}^{n}A_{ij}C_{ij}(A)$.

¶ For every $j\in[n]$, $\det A=\sum_{i=1}^{n}A_{ij}C_{ij}(A)$.

¶ $A\operatorname{adj}(A)=\operatorname{adj}(A)A=(\det A)I_{n}$.