Relation between a Determinant and its Cofactor Determinant

Let $|A|=\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}$, and let $A_1,B_1,C_1,\ldots,A_3,B_3,C_3$ be the cofactors of $a_1,b_1,c_1,\ldots,a_3,b_3,c_3$ respectively.

Two companion facts about cofactors:

1. Expanding a row (or column) against its own cofactors reproduces the determinant: $a_1A_1+b_1B_1+c_1C_1=|A|$ (this is just Laplace expansion, restated).
2. Expanding a row (or column) against the cofactors of a different row (column) always gives $0$: $a_1A_2+b_1B_2+c_1C_2=0$, and likewise for every other mismatched pairing (Note 7.12).

Now form the cofactor determinant $\begin{vmatrix}A_1&B_1&C_1\\A_2&B_2&C_2\\A_3&B_3&C_3\end{vmatrix}$ and multiply it (row-by-row, §7.3.4) with $|A|$ itself. Every diagonal product comes out $|A|$ (fact 1), and every off-diagonal product comes out $0$ (fact 2), so the product determinant is simply $|A|^3$ (the determinant of a diagonal matrix with $|A|$ repeated three times). Since the product of two determinants equals the determinant of the matrix product,

$|A|\times\begin{vmatrix}A_1&B_1&C_1\\A_2&B_2&C_2\\A_3&B_3&C_3\end{vmatrix}=|A|^3.$ …