Definition:Adjugate

Let $$D = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn}\end{vmatrix}$$ be a determinant of order $$n$$.

Let $$a_{rs}$$ be an element of $$D$$.

Let $$A_{rs}$$ be the cofactor of $$a_{rs}$$ in $$D$$.

Then the determinant $$D^* = \begin{vmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\ A_{21} & A_{22} & \cdots & A_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{n1} & A_{n2} & \cdots & A_{nn}\end{vmatrix}$$ is called the adjugate (determinant) of $$D$$.