Let A = 5 sin 2 θ cos 2 θ - sin 2 θ - 5 1 cos 2 θ 1 5 . Then maximum value of det A is

$A=\left[\begin{array}{ccc}5& {\sin }^2\theta & {\cos }^2\theta \\ -{\sin }^2\theta & -5& 1\\ {\cos }^2\theta & 1& 5\end{array}\right]$

$\left|A\right|=5\left(-26\right)-{\sin }^2\theta \left(-5{\sin }^2\theta +{\cos }^2\theta \right)+{\cos }^2\theta \left(-{\sin }^2\theta +5{\cos }^2\theta \right)$

$=-130+5{\sin }^4\theta -{\sin }^2\theta {\cos }^2\theta -{\sin }^2\theta {\cos }^2\theta +5{\cos }^4\theta$

$=-130+5\left({\sin }^4\theta +{\cos }^4\theta \right)-2{\sin }^2\theta {\cos }^2\theta$

$=-130+5\left[{\left({\sin }^2\theta +{\cos }^2\theta \right)}^2-2{\sin }^2\theta {\cos }^2\theta \right]-2{\sin }^2\theta {\cos }^2\theta$

$=-125-12{\sin }^2\theta {\cos }^2\theta$

$=-125-3{\left(2\sin \theta \cos \theta \right)}^2=-125-3{\sin }^{2}2\theta$

We know ${\sin }^{2}2\theta \in \left[0,1\right]$

Hence, ${\left|A\right|}_{max}=-125$