Prove that for a 3x3 matrix:
$$\varDelta^c=\varDelta^2$$ $$ \begin{bmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{bmatrix} = \begin{bmatrix} b_2c_3-c_2b_3 & c_2a_3-a_2c_3 & a_2b_3-b_2a_3 \\ a_2c_3-c_2a_3 & a_1c_3-a_3c_1 & a_1c_2-a_2c_1 \\ a_2b_3-a_3b_2 & a_1b_3-a_3b_1 & a_1b_2-a_2b_1 \end{bmatrix} = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}^2 $$
Trying to expand the left hand results in an unhandlable expression. I couldn't find this proof with a simple search either.