Minor, Cofactor, and Adjugate of a Real Square Matrix

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 $\mathbb{N}_{0}$, let $A\in\mathcal{M}_{n}(\mathbb{R})$ be a square real matrix with entries $A_{kl}$, and let $i,j\in[n]$, with the deletion index maps $s_{i},s_{j}:[n-1]\to[n]$, the determinant $\det$ on $\mathcal{M}_{n-1}(\mathbb{R})$, powers as in Commutative Rings, Fields and Ordered Fields: Standard Notation (iii), and the difference $n-1$ in $\mathbb{N}_{0}$ as in Commutative Rings, Fields and Ordered Fields: Standard Notation (iv), which lies in $\mathbb{N}$ by The Index Map Skipping One Element of [n] (i).

(i) The matrix obtained from $A$ by deleting row $i$ and column $j$ is the matrix $A^{(i,j)}\in\mathcal{M}_{n-1}(\mathbb{R})$ with $(A^{(i,j)})_{kl}=A_{s_{i}(k)s_{j}(l)}$ for all $k,l\in[n-1]$, which exists and is unique by The Determinant Map and the Matrices Formed from a Square Matrix by Reindexing Rows, Replacing a Row and Deleting a Row and a Column Exist and Are Unique (v).

(ii) The $(i,j)$ minor of $A$ is $M_{ij}(A)=\det A^{(i,j)}$.

(iii) The $(i,j)$ cofactor of $A$ is $C_{ij}(A)=(-1)^{i+j}M_{ij}(A)$.

(iv) The adjugate of $A$ is the matrix $\operatorname{adj}(A)\in\mathcal{M}_{n}(\mathbb{R})$ with $\operatorname{adj}(A)_{kl}=(-1)^{l+k}\det A^{(l,k)}$, with $A^{(l,k)}$ the matrix of clause deletion with $l,k$ in place of $i,j$, for all $k,l\in[n]$, which exists and is unique by The Determinant Map and the Matrices Formed from a Square Matrix by Reindexing Rows, Replacing a Row and Deleting a Row and a Column Exist and Are Unique (vi), $\det$ on $\mathcal{M}_{n-1}(\mathbb{R})$ being the map $D_{n-1}$ there by Determinant of a Real Square Matrix (i), so that, by clauses minor and cofactor with $j,i$ in place of $i,j$, $\operatorname{adj}(A)_{ij}=(-1)^{j+i}M_{ji}(A)=C_{ji}(A)$.