Minor, Cofactor, and Adjugate of a Real Square Matrix

Let $n$ be a natural number with $n\ge 2$, and let $A=(a_{ij})_{1\le i,j\le n}$ be an $n\times n$ real matrix.

For indices $i,j\in\{1,\dots,n\}$, the $(i,j)$ minor of $A$, denoted $M_{ij}(A)$, is the determinant of the $(n-1)\times(n-1)$ real matrix obtained from $A$ by deleting row $i$ and column $j$.

The $(i,j)$ cofactor of $A$ is

The adjugate matrix of $A$ (also called the classical adjoint) is the $n\times n$ real matrix

i.e., the matrix whose $(i,j)$ entry is the $(j,i)$ cofactor of $A$.