Jacobi's Theorem

Let $M_r$ be an $r$-rowed Minor of the $n$th order Determinant $\vert{\hbox{\sf A}}\vert$ associated with an $n\times n$ Matrix ${\hbox{\sf A}}=a_{ij}$ in which the rows $i_1$, $i_2$, ..., $i_r$ are represented with columns $k_1$, $k_2$, ..., $k_r$. Define the complementary minor to $M_r$ as the $(n-k)$-rowed Minor obtained from $\vert{\hbox{\sf A}}\vert$ by deleting all the rows and columns associated with $M_r$ and the signed complementary minor $M^{(r)}$ to $M_r$ to be

$$M^{(r)}=(-1)^{i_1+i_2+\ldots+i_r+k_1+k_2+\ldots+k_r}\times\hbox{[complementary minor to } M_r].$$

Let the Matrix of cofactors be given by

$$\Delta=\left\vert\matrix{ A_{11} & A_{12} & \cdots & A_{1n}\... ... & \vdots\cr A_{n1} & A_{n2} & \cdots & A_{nn}\cr}\right\vert,$$

with $M_r$ and $M_r'$ the corresponding $r$-rowed minors of $\vert{\hbox{\sf A}}\vert$ and $\Delta$, then it is true that

$$M_r'=\vert{\hbox{\sf A}}\vert^{r-1}M^{(r)}.$$

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1109-1100, 1979.