Adjugate Formula for the Matrix Inverse

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 $\det A\neq0$, with entries of matrices, the determinant $\det$, invertible matrices and their inverses $A^{-1}$, the cofactors $C_{ij}(A)$ and the adjugate $\operatorname{adj}(A)$, the scalar multiple of a matrix, and reciprocals and quotients as in Commutative Rings, Fields and Ordered Fields: Standard Notation (v).

(i) $A$ is invertible, and $A^{-1}=(1/\det A)\operatorname{adj}(A)$.

(ii) $(A^{-1})_{ij}=C_{ji}(A)/\det A$ for all $i,j\in[n]$, $A^{-1}$ existing by clause formula.