$\begin{aligned} & \Delta=\left|\begin{array}{cccc} a^2 & a b & a b & b^2 \\ a b & a^2 & b^2 & a b \\ a b & b^2 & a^2 & a b \\ b^2 & a b & a b & a^2 \end{array}\right|=\ell_2-\ell_3\left|\begin{array}{cccc} & \ell_4 \\ a^2-b^2 & 0 & 0 & b^2-a^2 \\ 0 & a^2-b^2 & b^2-a^2 & 0 \\ a b & b^2 & a^2 & a b \\ b^2 & a b & a b & a^2 \end{array}\right| \\ & =\left(a^2-b^2\right)^2 \cdot\left|\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ a b & b^2 & a^2 & a b \\ b^2 & a b & a b & a^2 \end{array}\right|=(a-b)^2 \cdot(a+b)^2 \cdot\left|\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ a b & b^2 & a^2 & a b \\ b^2 & a b & a b & a^2 \end{array}\right| \\ & c_1+c_2+c_3+c_4 \\ & =(a-b)^2 \cdot(a+b)^2 \cdot\left|\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ (a+b)^2 & b^2 & a^2 & a b \\ (a+b)^2 & a b & a b & a^2 \end{array}\right| \\ & =(a-b)^2 \cdot(a+b)^4 \cdot\left|\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ 1 & b^2 & a^2 & a b \\ 1 & a b & a b & a^2 \end{array}\right|=(a-b)^2 \cdot(a+b)^4 \cdot\left|\begin{array}{ccc} 0 & 1 & -1 \\ 1 & b^2 & a^2 \\ 1 & a b & a b \end{array}\right| \\ & \Delta^{\prime}=\left|\begin{array}{ccc} 0 & 1 & -1 \\ 1 & b^2 & a^2 \\ 1 & a b & a b \end{array}\right|=\left|\begin{array}{ccc} 0 & 1 & 0 \\ 1 & b^2 & a^2+b^2 \\ 1 & a b & 2 a b \end{array}\right|=-\left|\begin{array}{cc} 1 & a^2+b^2 \\ 1 & 2 a b \end{array}\right|=a^2+b^2-2 a b \\ & \Delta=(a-b)^4 \cdot(a+b)^4 \\ & \end{aligned}$

Calcul de déterminant