Let $A=\left[\begin{array}{ccc}1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1\end{array}\right] 0 \leq \theta \leq 2 \pi$, then :

$|A| \in[2,4]$

$|A|=\left|\begin{array}{ccc}1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1\end{array}\right|$

⇒ operation (R₃ → R₃ + R₁)

$|A|=\left|\begin{array}{ccc}1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ 0 & 0 & 2\end{array}\right|$

expanding across R₃ 

⇒ 2(1 + cos²θ) = |A|

= 0 ≤ θ ≤ 2π

⇒ 0 ≤ cos²θ ≤ 1

1 ≤ 1 + cos²θ ≤ 2

2 ≤ 2 (1 + cos²θ) ≤ 4  ⇒  |A| ∈ [2, 4]